Resolution
of
Nonsingularities,
Point-theoreticity,
and
Metric-admissibility
for
p-adic
Hyperbolic
Curves
Shinichi
Mochizuki
and
Shota
Tsujimura
June
16,
2023
Abstract
In
this
paper,
we
prove
that
arbitrary
hyperbolic
curves
over
p-adic
lo-
cal
fields
admit
resolution
of
nonsingularities
[“RNS”].
This
result
may
be
regarded
as
a
generalization
of
results
concerning
resolution
of
nonsingu-
larities
obtained
by
A.
Tamagawa
and
E.
Lepage.
Moreover,
by
combining
our
RNS
result
with
techniques
from
combinatorial
anabelian
geometry,
we
prove
that
an
absolute
version
of
the
geometrically
pro-Σ
Grothendieck
Conjecture
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields,
where
Σ
denotes
a
set
of
prime
numbers
of
cardinality
≥
2
that
contains
p,
holds.
This
settles
one
of
the
major
open
questions
in
anabelian
geometry.
Furthermore,
we
prove
—
again
by
applying
RNS
and
combinatorial
an-
abelian
geometry
—
that
the
various
p-adic
versions
of
the
Grothendieck-
Teichmüller
group
that
appear
in
the
literature
in
fact
coincide.
As
a
corollary,
we
conclude
that
the
metrized
Grothendieck-Teichmüller
group
is
commensurably
terminal
in
the
Grothendieck-Teichmüller
group.
This
settles
a
longstanding
open
question
in
combinatorial
anabelian
geometry.
Contents
Introduction
2
Notations
and
Conventions
11
1
Local
construction
of
Artin-Schreier
extensions
in
the
special
fiber
15
2020
Mathematics
Subject
Classification:
Primary
14H30;
Secondary
14H25.
Keywords
and
phrases:
anabelian
geometry;
resolution
of
nonsingularities;
abso-
lute
Grothendieck
Conjecture;
combinatorial
anabelian
geometry;
Grothendieck-Teichmüller
group;
étale
fundamental
group;
tempered
fundamental
group;
hyperbolic
curve;
configuration
space.
1
2
Resolution
of
nonsingularities
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields
26
3
Point-theoreticity,
metric-admissibility,
and
arithmetic
cuspi-
dalization
72
References
103
Introduction
Let
p
be
a
prime
number;
Σ
a
nonempty
subset
of
the
set
Primes
of
prime
numbers.
For
a
connected
noetherian
scheme
S,
we
shall
write
Π
S
for
the
étale
fundamental
group
of
S,
relative
to
a
suitable
choice
of
basepoint.
For
any
field
F
of
characteristic
0,
any
field
extension
F
⊆
E,
and
any
algebraic
variety
[i.e.,
a
separated,
geometrically
integral
scheme
of
finite
type]
Z
over
F
,
we
shall
def
write
Z
E
=
Z
×
F
E
and
denote
by
F
an
algebraic
closure
[well-defined
up
to
def
isomorphism]
of
F
and
by
G
F
=
Gal(F
/F
)
the
absolute
Galois
group
of
F
.
For
any
field
F
of
characteristic
0
and
any
algebraic
variety
Z
over
F
,
we
shall
write
(Σ)
def
Π
Z
=
Π
Z
/Ker(Π
Z
F
Π
Σ
Z
F
),
where
Π
Z
F
Π
Σ
Z
F
denotes
the
maximal
pro-Σ
quotient.
Here,
we
recall
that
(Σ)
Π
Z
is
often
referred
to
as
the
geometrically
pro-Σ
fundamental
group
of
Z.
We
shall
write
Q
p
for
the
field
of
p-adic
numbers;
C
p
for
the
p-adic
completion
of
Q
p
.
We
shall
refer
to
a
finite
extension
field
of
Q
p
as
a
p-adic
local
field.
For
any
hyperbolic
curve
Z
over
either
the
algebraic
closure
of
a
mixed
character-
istic
complete
discrete
valuation
field
of
residue
characteristic
p
or
the
p-adic
completion
of
such
an
algebraic
closure,
we
shall
write
Π
tp
Z
for
the
Σ-tempered
fundamental
group
of
Z,
relative
to
a
suitable
choice
of
base-
point
[cf.
the
subsection
in
Notations
and
Conventions
entitled
“Fundamental
groups”].
If
S
1
,
S
2
are
schemes,
then
we
shall
write
Isom(S
1
,
S
2
)
for
the
set
of
isomorphisms
of
schemes
between
S
1
and
S
2
.
If
G
1
,
G
2
are
profinite
groups,
then
we
shall
write
OutIsom(G
1
,
G
2
)
for
the
set
of
isomorphisms
of
profinite
groups,
considered
up
to
composition
with
an
inner
automorphism
arising
from
an
element
∈
G
2
.
In
the
present
paper,
we
give
a
complete
affirmative
answer
to
the
following
question:
2
Does
an
arbitrary
hyperbolic
curve
over
a
p-adic
local
field
admit
resolution
of
nonsingularities?
Before
continuing,
we
review
the
notion
of
resolution
of
nonsingularities.
Let
K
(⊆
K)
be
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
v
a
valuation
on
a
field
F
that
contains
K.
Write
O
K
for
the
ring
of
integers
of
K;
O
v
for
the
ring
of
integers
determined
by
v;
m
v
⊆
O
v
for
the
maximal
ideal
of
O
v
.
Then
we
shall
say
that
v
is
a
p-valuation
[over
K]
[cf.
Definition
2.2,
(i)]
if
O
K
=
O
v
∩
K
[which
implies
that
p
∈
m
v
].
Here,
the
phrase
“over
K”
will
be
omitted
in
situations
where
the
base
field
K
is
fixed
throughout
the
discussion.
We
shall
say
that
v
is
residue-transcendental
def
[cf.
Definition
2.2,
(i)]
if
it
is
a
p-valuation
whose
residue
field
k
v
=
O
v
/m
v
is
a
transcendental
extension
of
the
residue
field
of
K.
Let
Z
be
a
hyperbolic
curve
over
K.
Then
we
shall
say
that
Z
satisfies
Σ-RNS
[i.e.,
“Σ-resolution
of
nonsingularities”
—
cf.
Definition
2.2,
(vii);
[Lpg1],
Definition
2.1]
if
the
following
condition
holds:
Let
v
be
a
discrete
residue-transcendental
p-valuation
on
the
func-
tion
field
K(Z)
of
Z.
Then
there
exists
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
Y
→
Z
such
that
Y
has
stable
reduction
[over
its
base
field],
and
v
coincides
with
the
restriction
[to
K(Z)]
of
a
discrete
valuation
on
the
function
field
K(Y
)
of
Y
that
arises
from
an
irreducible
component
of
the
special
fiber
of
the
stable
model
[cf.
Definition
2.1,
(vi)]
of
Y
.
If
Z
is
an
O
K
-scheme,
then
we
shall
write
Z
s
for
the
special
fiber
of
Z
[i.e.,
the
fiber
of
Z
over
the
closed
point
of
Spec
O
K
].
Let
Z
be
an
O
K
-scheme.
Then
[cf.
Definition
2.1,
(i),
(ii),
(iv)]:
(i)
We
shall
say
that
Z
is
a
compactified
model
of
Z
over
O
K
if
Z
is
a
proper,
flat,
normal
scheme
over
O
K
whose
generic
fiber
is
the
[uniquely
deter-
mined,
up
to
unique
isomorphism]
smooth
compactification
of
Z
over
K.
(ii)
Suppose
that
the
cusps
of
Z
are
K-rational.
Then
we
shall
say
that
Z
is
a
compactified
stable
model
of
Z
over
O
K
if
Z
is
a
compactified
model
of
Z
over
O
K
such
that
the
following
conditions
hold:
•
the
geometric
special
fiber
of
Z
is
a
semistable
curve
[i.e.,
a
reduced,
connected
curve
each
of
whose
nonsmooth
points
is
an
ordinary
dou-
ble
point];
•
the
images
of
the
sections
Spec
O
K
→
Z
determined
by
the
cusps
of
Z
[which
we
shall
refer
to
as
cusps
of
Z]
lie
in
the
smooth
locus
of
Z
and
do
not
intersect
each
other.
•
Z,
together
with
the
cusps
of
Z,
determines
a
pointed
stable
curve.
Then
one
verifies
immediately,
by
considering
blow-ups
of
compactified
models
at
specified
closed
points
[cf.
also
Remark
2.2.2;
Proposition
2.4,
(iii),
(iv)],
that,
if
Σ
is
a
set
of
cardinality
≥
2
that
contains
p
[cf.
the
situation
discussed
3
in
Theorem
A
below],
then
the
condition
of
satisfying
Σ-RNS
discussed
above
is
in
fact
equivalent
to
the
following
property,
which
clarifies
the
meaning
of
“resolution
of
nonsingularities”:
Let
Z
be
a
compactified
model
of
Z
over
O
K
;
z
∈
Z
s
a
closed
point.
Then,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exist
•
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
Y
→
Z
of
hyperbolic
curves
over
K,
•
a
compactified
stable
model
Y
of
Y
over
O
K
,
•
a
morphism
Y
→
Z
of
compactified
models
over
O
K
that
re-
stricts
to
the
finite
étale
Galois
covering
Y
→
Z,
•
an
irreducible
component
D
of
Y
s
whose
normalization
is
of
genus
≥
1,
and
whose
image
in
Z
s
is
z
∈
Z
s
.
This
alternative
formulation
of
the
condition
of
satisfying
Σ-RNS
is
useful
to
keep
in
mind
when
considering
the
relationship
between
Theorem
A
below
and
the
following
result
due
to
A.
Tamagawa
[cf.
[Tama2]],
which
played
an
important
role
in
motivating
the
following
result
due
to
E.
Lepage
[cf.
[Lpg1]]:
•
Suppose
that
Σ
=
Primes,
that
the
residue
field
of
K
is
algebraic
over
the
finite
field
of
cardinality
p,
and
that
Z
is
the
compactified
stable
model
of
Z
over
O
K
.
Then
there
exist
a
connected
finite
étale
Galois
covering
Y
→
Z
and
an
irreducible
component
D
of
Y
s
as
in
the
above
alternative
formulation
[cf.
[Tama2],
Theorem
0.2,
(v)].
•
Suppose
that
Σ
=
Primes,
that
K
is
a
p-adic
local
field,
and
that
Z
is
a
hyperbolic
Mumford
curve
over
K.
Then
Z
satisfies
Σ-RNS
[cf.
[Lpg1],
Theorem
2.7].
Our
first
main
result
may
be
regarded
as
a
generalization
of
these
results
[cf.
Theorem
2.17]:
Theorem
A
(Resolution
of
nonsingularities
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields).
Suppose
that
Σ
⊆
Primes
is
a
subset
of
cardinality
≥
2
that
contains
p,
and
that
K
is
a
p-adic
local
field.
Let
X
be
a
hyperbolic
curve
over
K;
L
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p
that
contains
K
as
a
topological
subfield.
Then
X
L
satisfies
Σ-RNS
if
and
only
if
the
residue
field
of
L
is
algebraic
over
the
finite
field
of
cardinality
p.
In
the
remainder
of
the
present
Introduction,
we
discuss
various
anabelian
applications/consequences
of
Theorem
A.
First,
by
applying
a
certain
sophisticated
version
of
the
argument
applied
in
the
proof
of
[Tsjm],
Theorem
2.2,
we
obtain
the
following
consequence
of
Theorem
A
concerning
the
determination
of
closed
points
on
arbitrary
hyperbolic
curves
via
geometric
tempered
fundamental
groups,
which
generalizes
[Tsjm],
Theorem
2.2
[cf.
Corollary
2.5,
(ii),
which
in
fact
applies
to
hyperbolic
curves
over
more
general
p-adic
fields;
Remark
2.5.1]:
4
Theorem
B
(Determination
of
closed
points
on
arbitrary
p-adic
hy-
perbolic
curves
by
geometric
tempered
fundamental
groups).
Suppose
that
Σ
⊆
Primes
is
a
subset
of
cardinality
≥
2
that
contains
p.
Let
X
†
,
X
‡
†
→
X
†
,
X
‡
→
X
‡
for
the
univer-
be
hyperbolic
curves
over
Q
p
.
Write
X
,
Π
tp
,
respectively.
Let
sal
geometrically
pro-Σ
coverings
corresponding
to
Π
tp
X
†
X
‡
x
†
∈
X
†
(C
p
),
x
‡
∈
X
‡
(C
p
).
Write
X
x
†
†
(respectively,
X
x
‡
‡
)
for
the
hyperbolic
curve
X
C
†
p
\{x
†
}
(respectively,
X
C
‡
p
\{x
‡
})
over
C
p
.
Let
∼
σ
:
Π
tp
†
→
Π
tp
‡
x
x
X
†
X
‡
be
an
isomorphism
of
topological
groups
that
fits
into
a
commutative
diagram
∼
Π
tp
†
−−−−→
Π
tp
‡
X
†
X
‡
σ
⏐
x
⏐
x
⏐
⏐
∼
Π
tp
−−−−→
Π
tp
,
X
†
X
‡
σ
where
the
vertical
arrows
are
the
natural
surjections
[determined
up
to
com-
position
with
an
inner
automorphism]
induced
by
the
natural
open
immersions
X
x
†
†
→
X
C
†
p
,
X
x
‡
‡
→
X
C
‡
p
of
hyperbolic
curves;
the
lower
horizontal
arrow
σ
is
the
isomorphism
of
topological
groups
[determined
up
to
composition
with
an
inner
automorphism]
induced
by
a(n)
[uniquely
determined]
isomorphism
∼
σ
X
:
X
†
→
X
‡
of
schemes
over
Q
p
.
Then
x
‡
=
σ
X
(x
†
).
Next,
we
consider
applications
of
Theorem
A
to
Grothendieck
Conjecture-
type
results
in
anabelian
geometry.
We
begin
by
recalling
the
following
question,
which
may
may
be
considered
as
an
absolute
version
of
the
Grothendieck
Con-
jecture
for
hyperbolic
curves
over
p-adic
local
fields:
Let
X
†
,
X
‡
be
hyperbolic
curves
over
p-adic
local
fields.
Then
is
the
natural
map
Isom(X
†
,
X
‡
)
−→
OutIsom(Π
X
†
,
Π
X
‡
)
bijective?
This
question
may
be
regarded
as
one
of
the
major
open
questions
in
anabelian
geometry.
In
this
context,
we
recall
that,
in
the
case
of
the
relative
version
of
the
Grothendieck
Conjecture
for
arbitrary
hyperbolic
curves,
many
satisfactory
results
have
been
obtained
[cf.
[PrfGC],
Theorem
A;
[Tama1],
Theorem
0.4;
[LocAn],
Theorem
A].
In
particular,
the
first
author
of
the
present
paper
gave
a
complete
affirmative
answer
to
the
original
question
posed
by
A.
Grothendieck
[i.e.,
the
original
“Grothendieck
Conjecture”]
in
quite
substantial
generality
[cf.
[LocAn],
Theorem
A;
[AnabTop],
Theorem
4.12].
On
the
other
hand,
in
the
case
5
of
the
absolute
version
of
the
Grothendieck
Conjecture
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields
[i.e.,
the
question
of
the
above
display],
analogous
results
had
not
been
obtained
previously,
due
to
the
existence
of
outer
isomor-
phisms
of
the
absolute
Galois
groups
of
p-adic
local
fields
that
do
not
arise
from
isomorphisms
of
fields
[cf.,
e.g.,
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7].
In
this
direction,
in
some
sense
the
strongest
known
result,
prior
to
the
present
paper,
was
the
following
result
[cf.
[AbsTopII],
Corollary
2.9]:
Suppose
that
Σ
⊆
Primes
is
a
subset
of
cardinality
≥
2
that
contains
p.
Let
X
†
,
X
‡
be
hyperbolic
curves
over
p-adic
local
fields.
Write
(Σ)
(Σ)
(Σ)
(Σ)
OutIsom
D
(Π
X
†
,
Π
X
‡
)
⊆
OutIsom(Π
X
†
,
Π
X
‡
)
for
the
subset
determined
by
the
isomorphisms
that
induce
bijections
between
the
respective
sets
of
decomposition
subgroups
associated
to
the
closed
points
of
X
†
and
X
‡
.
Then
the
natural
map
Isom(X
†
,
X
‡
)
−→
OutIsom
D
(Π
X
†
,
Π
X
‡
)
(Σ)
(Σ)
is
bijective.
Moreover,
in
[AbsTopII],
[AbsTopIII],
the
first
author
developed
a
technique,
called
“Belyi
cuspidalization”,
that
allows
one
to
reconstruct
the
decomposition
subgroups
associated
to
the
closed
points
of
strictly
Belyi-type
hyperbolic
curves
and
proved
that
an
absolute
version
of
the
Grothendieck
Conjecture
holds
for
such
curves
[cf.
[AbsTopIII],
Theorem
1.9].
In
the
present
paper,
we
apply
Theorem
A,
together
with
some
combinatorial
anabelian
geometry,
to
reconstruct
the
set
of
C
p
-valued
points
of
a
hyperbolic
curve
over
Q
p
from
its
geometric
tempered
fundamental
group
[cf.
Corollary
3.10,
which
in
fact
applies
to
hyperbolic
curves
over
more
general
p-adic
fields]:
Theorem
C
(Reconstruction
of
C
p
-valued
points
via
geometric
tem-
pered
fundamental
groups).
Suppose
that
Σ
⊆
Primes
is
a
subset
of
cardi-
→
X
nality
≥
2
that
contains
p.
Let
X
be
a
hyperbolic
curve
over
Q
p
.
Write
X
tp
may
be
for
the
universal
pro-Σ
covering
corresponding
to
Π
X
[so
Gal(
X/X)
tp
identified
with
the
pro-Σ
completion
of
Π
X
].
Then
the
set
X(C
p
)
equipped
—
hence
also,
by
passing
to
the
set
of
with
its
natural
action
by
Gal(
X/X)
p
)
X(C
p
)
—
may
be
reconstructed,
Gal(
X/X)-orbits,
the
quotient
set
X(C
in
a
purely
combinatorial/group-theoretic
way
and
functorially
with
respect
to
isomorphisms
of
topological
groups,
from
the
underlying
topological
group
of
Π
tp
X
.
Note
that
it
follows
immediately
from
Theorem
C
that,
under
the
assump-
tion
that
Σ
⊆
Primes
is
a
subset
of
cardinality
≥
2
that
contains
p,
one
may
reconstruct
the
set
of
closed
points,
hence
also
the
set
of
associated
de-
composition
subgroups,
of
a
hyperbolic
curve
over
a
p-adic
local
field,
in
a
purely
combinatorial/group-theoretic
way
and
functorially
with
respect
to
iso-
morphisms
of
topological
groups,
from
its
geometrically
pro-Σ
étale
fundamental
group
[cf.
the
proof
of
Theorem
3.11
for
more
details].
Here,
it
is
also
interesting
to
observe
that
6
this
reconstruction
of
the
set
of
decomposition
subgroups
asso-
ciated
to
closed
points
from
the
geometrically
pro-Σ
étale
funda-
mental
group
of
a
hyperbolic
curve
over
a
p-adic
local
field
may
be
thought
of
as
a
sort
of
a
weak
version
of
the
Section
Conjecture.
In
particular,
by
applying
[AbsTopII],
Corollary
2.9,
together
with
some
com-
binatorial
anabelian
geometry,
we
obtain
a
complete
affirmative
answer
to
the
question
considered
above,
i.e.,
an
absolute
version
of
the
Grothendieck
Con-
jecture
for
arbitrary
hyperbolic
curves,
as
well
as
for
the
configuration
spaces
associated
to
such
hyperbolic
curves,
over
p-adic
local
fields
[cf.
Theorems
3.12;
3.13]:
Theorem
D
(Absolute
version
of
the
Grothendieck
Conjecture
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields).
Suppose
that
Σ
⊆
Primes
is
a
subset
of
cardinality
≥
2
that
contains
p.
Let
X
†
,
X
‡
be
hyperbolic
curves
over
p-adic
local
fields.
Then
the
natural
map
Isom(X
†
,
X
‡
)
−→
OutIsom(Π
X
†
,
Π
X
‡
)
(Σ)
(Σ)
is
bijective.
Theorem
E
(Absolute
version
of
the
Grothendieck
Conjecture
for
con-
figuration
spaces
associated
to
arbitrary
hyperbolic
curves
over
p-adic
local
fields).
Let
X
†
,
X
‡
be
hyperbolic
curves
over
p-adic
local
fields;
n
†
,
n
‡
positive
integers.
Write
X
n
†
†
(respectively,
X
n
‡
‡
)
for
the
n
†
-th
(respectively,
n
‡
-
th)
configuration
space
associated
to
X
†
(respectively,
X
‡
).
Then
the
natural
map
Isom(X
n
†
†
,
X
n
‡
‡
)
−→
OutIsom(Π
X
†
,
Π
X
‡
)
n
†
n
‡
is
bijective.
In
the
context
of
the
relationship
between
the
theory
developed
in
the
present
paper
and
the
Section
Conjecture,
it
is
of
interest
to
note
the
following
conse-
quences
of
the
theory
underlying
Theorem
C
[cf.
Proposition
2.4,
(vii);
Propo-
sition
3.5,
(iii);
Proposition
3.9,
(iv)
—
all
of
which
in
fact
apply
to
hyperbolic
curves
over
more
general
p-adic
fields]:
Theorem
F
(Consequences
related
to
the
Section
Conjecture
over
p-adic
fields).
Suppose
that
Σ
⊆
Primes
is
a
subset
that
contains
p,
and
that
the
residue
field
of
K
is
an
algebraic
extension
of
the
finite
field
of
cardinality
p.
Let
l
∈
Σ
\
{p};
H
⊆
G
K
a
closed
subgroup
such
that
the
intersection
H
∩
I
K
of
H
with
the
inertia
subgroup
I
K
of
G
K
admits
a
surjection
to
[the
profinite
group]
Z
l
.
Write
Ω
for
the
p-adic
completion
of
K;
Ω
H
⊆
Ω
for
the
subfield
of
Ω
fixed
by
H.
Then
the
following
properties
hold:
(i)
Let
X
be
a
proper
hyperbolic
curve
over
K
that
is
in
fact
defined
over
a
p-adic
local
field
contained
[in
a
fashion
compatible
with
the
respective
→
X
a
universal
geometrically
pro-Σ
covering,
s
X
:
topologies]
in
K,
X
7
(Σ)
def
H
→
Π
X
=
Gal(
X/X)
a
section
of
the
restriction
to
H
of
the
natural
(Σ)
an
for
the
topological
pro-Berkovich
space
surjection
Π
X
G
K
.
Write
X
associated
to
[i.e.,
the
inverse
limit
of
the
underlying
topological
spaces
Then
of
the
Berkovich
spaces
associated
to
the
finite
subcoverings
of
]
X.
an
there
exists
at
most
one
point
∈
X
that
is
fixed
by
the
restriction,
via
(Σ)
s
X
,
to
H
of
the
natural
action
of
Π
X
on
the
topological
pro-Berkovich
an
;
if,
moreover,
the
restriction
to
H
of
the
l-adic
cyclotomic
space
X
character
of
K
has
open
image,
then
there
exists
a
unique
such
point
an
.
Finally,
s
X
arises
from
an
Ω
H
-rational
point
∈
X(Ω
H
)
if
and
∈
X
(Σ)
an
.
only
if
its
image
is
a
maximal
stabilizer
⊆
Π
X
×
G
K
H
of
a
point
∈
X
(ii)
Suppose
that
the
restriction
to
H
of
the
l-adic
cyclotomic
character
of
K
has
open
image.
Let
Y
,
Z
be
[not
necessarily
proper!]
hyperbolic
curves
over
K
that
are
in
fact
defined
over
a
p-adic
local
field
contained
[in
a
fashion
compatible
with
the
respective
topologies]
in
K;
Y
→
Y
,
Z
→
Z
universal
geometrically
pro-Σ
coverings
of
Y
,
Z,
respectively;
(Σ)
def
f
:
Y
→
Z
a
dominant
morphism
over
K;
s
Y
:
H
→
Π
Y
=
Gal(
Y
/Y
),
(Σ)
def
=
Gal(
X/X)
sections
of
the
restrictions
to
H
of
the
s
Z
:
H
→
Π
Z
(Σ)
(Σ)
respective
natural
surjections
Π
Y
G
K
,
Π
Z
G
K
such
that
s
Y
is
(Σ)
mapped,
up
to
Π
Z
-conjugation,
by
f
to
s
Z
via
the
map
induced
by
f
on
geometrically
pro-Σ
fundamental
groups.
Then
s
Y
arises
from
an
Ω
H
-
rational
point
∈
Y
(Ω
H
)
if
and
only
if
s
Z
arises
from
an
Ω
H
-rational
point
∈
Z(Ω
H
).
Here,
we
observe
in
passing
that
it
follows
immediately
from
Theorem
F,
(ii)
[i.e.,
as
stated
above],
together
with
[BSC],
Theorem
5.33,
that
a
similar
result
to
the
result
stated
in
Theorem
F,
(ii),
holds
if
the
field
K
is
replaced
by
a
number
field,
but
[since
global
issues
over
number
fields
lie
beyond
the
scope
of
the
present
paper]
we
shall
not
discuss
this
in
detail
in
the
present
paper.
Next,
we
recall
that
one
of
the
key
ingredients
in
the
theory
of
p-adic
arith-
metic
cuspidalizations
developed
in
[Tsjm],
§2,
is
Lepage’s
resolution
of
nonsin-
gularities
[i.e.,
[Lpg1],
Theorem
2.7],
which
may
be
regarded
as
a
special
case
of
Theorem
A.
Our
next
result
is
obtained
by
applying
this
theory
of
p-adic
arith-
metic
cuspidalizations
[i.e.,
[Tsjm],
§2],
together
with
some
elementary
observa-
tions
concerning
the
lengths
of
nodes
of
stable
models
of
hyperbolic
curves
[cf.
Proposition
3.15],
to
prove
that
the
various
p-adic
versions
of
the
Grothendieck-
Teichmüller
group
that
appear
in
the
literature
[cf.
[Tsjm],
Remark
2.1.2]
in
fact
coincide
and
are
commensurably
terminal
in
the
Grothendieck-Teichmüller
group
[cf.
Theorem
3.16;
Corollary
3.17]:
Theorem
G
(Equality
and
commensurable
terminality
of
various
p-adic
versions
of
the
Grothendieck-Teichmüller
group).
Suppose
that
Σ
=
def
Primes.
Write
X
=
P
1
Q
\
{0,
1,
∞};
p
GT
⊆
Out(Π
X
)
8
for
the
Grothendieck-Teichmüller
group
[cf.
[CmbCsp],
Remark
1.11.1];
GT
M
⊆
GT
(⊆
Out(Π
X
))
for
the
metrized
Grothendieck-Teichmüller
group
[cf.
[CbTpIII],
Remark
3.19.2];
def
=
GT
∩
Out(Π
tp
GT
tp
p
X
)
⊆
Out(Π
X
)
[cf.
the
subsection
in
Notations
and
Conventions
entitled
“Fundamental
groups”;
[Tsjm],
Definition
2.1].
Then
the
natural
inclusion
GT
M
⊆
GT
tp
p
of
subgroups
of
GT
is
an
equality.
In
particular,
it
holds
that
GT
M
=
GT
p
=
GT
G
=
GT
tp
p
[cf.
[Tsjm],
Remark
2.1.2].
Moreover,
GT
M
=
GT
p
=
GT
G
=
GT
tp
p
is
com-
M
mensurably
terminal
in
GT,
i.e.,
the
commensurator
C
GT
(GT
)
of
GT
M
in
GT
is
equal
to
GT
M
.
Note
that
the
commensurable
terminality
in
the
final
portion
of
Theorem
G
may
be
regarded
as
an
affirmative
answer
to
the
question
posed
in
the
discussion
immediately
preceding
Theorem
E
in
[CbTpIII],
Introduction.
Finally,
we
conclude
with
an
interesting
complement
to
the
theory
of
p-adic
arithmetic
cuspidalizations
by
applying
Theorem
C,
together
with
the
theory
of
metric-admissibility
developed
in
[CbTpIII],
§3,
to
construct
certain
p-adic
arithmetic
cuspidalizations
of
the
geometric
tempered
fundamental
group
of
a
hyperbolic
curve
over
Q
p
equipped
with
certain
relatively
mild
auxiliary
data
[cf.
Definition
3.19;
Theorem
3.20;
Remarks
3.20.1,
3.20.2].
The
contents
of
the
present
paper
may
be
summarized
as
follows:
In
§1,
we
discuss
in
detail
certain
local
computations
—
motivated
by
[Lpg1],
Proposition
2.4,
but
formulated
entirely
in
the
language
of
schemes
and
for-
mal
schemes,
i.e.,
without
resorting
to
the
use
of
notions
from
the
theory
of
Berkovich
spaces
—
concerning
iterates
of
the
p-th
power
morphism
of
the
mul-
tiplicative
group
scheme
G
m
over
the
ring
of
integers
of
a
mixed
characteristic
discrete
valuation
field
of
residue
characteristic
p.
By
restricting
such
mor-
phisms
to
suitable
formal
neighborhoods,
we
conclude
that
smooth
curves
of
genus
≥
1
appear
in
the
special
fibers
of
suitable
models
of
the
domain
curves
of
such
morphisms,
i.e.,
as
certain
Artin-Schreier
coverings
of
curves
of
genus
0
[cf.
Proposition
1.6;
Remark
1.6.2].
In
§2,
we
begin
by
discussing
various
generalities
concerning
models
of
a
hyperbolic
curve
over
a
mixed
characteristic
complete
discrete
valuation
field
[cf.
Definition
2.1;
Proposition
2.3].
In
particular,
we
discuss
the
definition
of
the
notion
of
Σ-RNS
[cf.
Definition
2.2,
(vii)],
together
with
closely
related
basic
properties
of
this
notion
[cf.
Propositions
2.4;
Corollary
2.5].
Another
important
9
notion
in
this
context
is
the
purely
combinatorial
notion
of
VE-chains
associated
to
a
hyperbolic
curve
over
a
mixed
characteristic
complete
discrete
valuation
field
[cf.
Definition
2.2,
(iii)].
This
notion
is
closely
related
to
the
topological
Berkovich
space
associated
to
the
hyperbolic
curve
[cf.
Proposition
2.3,
(vii),
(viii)].
Finally,
we
apply
certain
constructions
involving
p-divisible
groups
to
extend
the
Artin-Schreier
coverings
constructed
locally
in
§1
to
coverings
of
an
arbitrary
hyperbolic
curve
over
a
p-adic
local
field
[cf.
Propositions
2.12,
2.13;
Theorem
2.16].
This
leads
naturally
to
a
proof
of
Theorem
A
[cf.
Theorem
2.17],
hence
also,
by
combining
Theorem
A
with
the
theory
of
VE-chains
developed
in
the
earlier
portion
of
§2,
together
with
some
combinatorial
anabelian
geometry,
of
Theorem
B.
In
§3,
we
begin
by
recalling
the
well-known
classification
of
the
points
of
the
topological
Berkovich
space
associated
to
a
proper
hyperbolic
curve
over
a
mixed
characteristic
complete
discrete
valuation
field
via
the
notion
of
type
i
points,
where
i
∈
{1,
2,
3,
4}.
Next,
we
introduce
a
certain
combinatorial
clas-
sification
of
the
VE-chains
considered
in
§2
and
observe
that
this
classification
of
VE-chains
leads
naturally
to
a
purely
combinatorial
characterization
of
the
well-known
classification
via
type
i
points
mentioned
above.
We
then
apply
the
theory
of
§2
to
give
a
group-theoretic
characterization,
motivated
by
[but
by
no
means
identical
to]
the
characterization
of
[Lpg2],
§4,
of
the
type
i
points
in
terms
of
the
geometric
Σ-tempered
fundamental
group
of
the
hyperbolic
curve.
The
theory
surrounding
this
group-theoretic
characterization
leads
naturally
to
proofs
of
Theorems
C
and
F.
Moreover,
by
combining
this
group-theoretic
characterization
with
[AbsTopII],
Corollary
2.9;
[HMM],
Theorem
A,
we
obtain
proofs
of
Theorems
D
and
E
[cf.
Theorems
3.12;
3.13].
We
then
switch
gears
to
discuss
metric-admissibility
for
p-adic
hyperbolic
curves.
This
discussion
of
metric-admissibility
yields,
in
particular,
a
proof
of
Theorem
G
and
leads
naturally
to
the
discussion
of
p-adic
arithmetic
cuspidalizations
associated
to
geometric
tempered
fundamental
groups
equipped
with
certain
relatively
mild
auxiliary
data
[cf.
Definition
3.19;
Theorem
3.20;
Remarks
3.20.1,
3.20.2]
men-
tioned
above.
Acknowledgements
This
research
was
supported
by
the
Research
Institute
for
Mathematical
Sci-
ences,
an
International
Joint
Usage/Research
Center
located
in
Kyoto
Univer-
sity,
as
well
as
the
International
Center
for
Research
in
Next
Generation
Geom-
etry.
10
Notations
and
Conventions
Numbers:
The
notation
Primes
will
be
used
to
denote
the
set
of
prime
numbers.
The
notation
N
will
be
used
to
denote
the
set
of
nonnegative
integers.
The
notation
Q
≥0
will
be
used
to
denote
the
additive
monoid
of
nonnegative
rational
numbers.
The
notation
Z
will
be
used
to
denote
the
additive
group
or
ring
of
integers.
The
notation
Q
will
be
used
to
denote
the
field
of
rational
numbers.
The
notation
R
will
be
used
to
denote
the
field
of
real
numbers.
For
each
x
∈
R,
the
notation
x
will
be
used
to
denote
the
greatest
integer
≤
x.
If
p
is
a
prime
number,
then
the
notation
Q
p
will
be
used
to
denote
the
field
of
p-adic
numbers;
the
notation
Z
p
will
be
used
to
denote
the
additive
group
or
ring
of
p-adic
integers;
the
notation
Q
p
will
be
used
to
denote
an
algebraic
closure
of
Q
p
;
the
notation
C
p
will
be
used
to
denote
the
p-adic
completion
of
Q
p
.
We
shall
refer
to
a
finite
extension
field
of
Q
p
as
a
p-adic
local
field.
Monoids:
Let
M
be
a
commutative
monoid.
In
this
subsection,
we
regard
the
set
of
positive
integers
N
≥1
as
a
directed
set
via
its
multiplicative
structure,
i.e.,
def
i
≤
j
⇔
i
|
j.
For
i
∈
N
≥1
,
write
M
i
=
M
.
For
i,
j
∈
N
≥1
such
that
i
≤
j,
write
φ
i,j
:
M
i
→
M
j
for
the
homomorphism
of
monoids
obtained
by
multiplication
by
ji
.
Then
we
shall
refer
to
the
inductive
limit
of
the
inductive
system
{M
i
,
φ
i,j
}
on
the
directed
set
N
≥1
as
the
perfection
of
M
.
Rings
and
fields:
Let
R
be
a
ring.
Then
we
shall
write
R
×
for
the
multiplicative
group
of
units
of
the
ring.
Let
F
be
a
perfect
field,
p
a
prime
number.
Then
the
notation
F
will
be
def
used
to
denote
an
algebraic
closure
of
F
;
G
F
=
Gal(F
/F
).
Suppose
that
F
is
a
valuation
field,
i.e.,
a
field
equipped
with
a
valuation
map
[cf.,
e.g.,
the
axioms
of
[EP],
Eq.
(2.1.2)].
Then
we
shall
write
O
F
for
the
ring
of
integers
of
F
;
m
F
for
the
maximal
ideal
of
O
F
.
Thus,
the
valuation
map
on
F
induces
an
isomorphism
of
ordered
abelian
groups
between
F
×
/O
F
×
and
the
value
group
of
the
valuation
field
F
.
In
particular,
any
valuation
map
on
the
field
F
is
determined,
up
to
unique
isomorphism
[in
the
evident
sense],
by
the
ring
of
integers
⊆
F
associated
to
the
valuation
map.
In
the
present
paper,
we
shall
use
the
term
valuation
to
refer
to
an
isomorphism
class
of
valuation
maps
on
a
field
F
,
i.e.,
the
collection
of
valuation
maps
that
give
rise
to
the
same
ring
of
integers
⊆
F
.
We
shall
refer
to
a
specific
valuation
map
within
a
given
isomorphism
class
of
valuation
maps
on
a
field
as
a
normalized
valuation,
i.e.,
an
isomorphism
class
of
valuation
maps
on
the
field,
together
with
a
specific
valuation
map
[belonging
to
this
class],
which
we
shall
refer
to
as
a
normalization.
If
the
value
group
of
the
11
valuation
field
F
is
isomorphic,
as
an
ordered
abelian
group,
to
a
subgroup
of
the
underlying
additive
group
of
R,
then
we
shall
say
that
F
is
a
real
valuation
field.
If
F
is
a
mixed
characteristic
real
valuation
field
of
residue
characteristic
p,
then
the
notation
v
p
will
be
used
to
denote
the
normalized
valuation
on
F
whose
normalization
is
determined
by
the
condition
that
v
p
(p)
=
1
∈
R.
If
F
is
a
henselian
valuation
field
of
characteristic
0,
then
we
shall
write
I
F
⊆
G
F
for
the
inertia
subgroup;
F
⊆
F
ur
(⊆
F
)
for
the
maximal
unramified
extension.
If
F
is
a
real
henselian
valuation
field
of
characteristic
0,
then
we
shall
write
F
ur
for
the
completion
of
F
ur
[cf.
Remark
2.2.4
for
more
details].
Topological
groups:
Let
G
be
a
topological
group;
H
⊆
G
a
closed
subgroup
of
G.
Then
we
shall
write
G
ab
for
the
abelianization
of
G;
C
G
(H)
for
the
commensurator
of
H
⊆
G,
i.e.,
def
C
G
(H)
=
{g
∈
G
|
H
∩
g
·
H
·
g
−1
is
of
finite
index
in
H
and
g
·
H
·
g
−1
}.
We
shall
say
that
the
closed
subgroup
H
is
commensurably
terminal
in
G
if
H
=
C
G
(H).
Let
Σ
⊆
Primes
be
a
nonempty
subset.
Then
we
shall
write
G
Σ
for
the
pro-Σ
completion
of
G.
We
shall
write
Aut(G)
for
the
group
of
continuous
automorphisms
of
the
topological
group
G,
Inn(G)
⊆
Aut(G)
for
the
subgroup
of
inner
automorphisms
def
of
G,
and
Out(G)
=
Aut(G)/Inn(G).
Suppose
that
G
is
center-free.
Then
we
have
a
natural
exact
sequence
of
groups
∼
1
−→
G
(
→
Inn(G))
−→
Aut(G)
−→
Out(G)
−→
1.
If
J
is
a
group,
and
ρ
:
J
→
Out(G)
is
a
homomorphism,
then
we
shall
denote
by
out
G
J
the
group
obtained
by
pulling
back
the
above
exact
sequence
of
groups
via
ρ.
Thus,
we
have
a
natural
exact
sequence
of
groups
out
1
−→
G
−→
G
J
−→
J
−→
1.
Suppose
further
that
the
topology
of
G
admits
a
countable
basis
consisting
of
characteristic
open
subgroups
of
G.
Then
one
verifies
immediately
that
the
topology
of
G
induces
a
natural
topology
on
the
group
Aut(G),
hence
on
the
group
Out(G).
In
particular,
one
verifies
easily
that
if
J
is
a
topological
group,
out
and
ρ
:
J
→
Out(G)
is
continuous,
then
G
J
admits
a
natural
topological
group
structure.
Let
G
1
,
G
2
be
profinite
groups.
Then
we
shall
write
OutIsom(G
1
,
G
2
)
12
for
the
set
of
isomorphisms
of
profinite
groups,
considered
up
to
composition
with
an
inner
automorphism
arising
from
an
element
∈
G
2
.
Semi-graphs:
Let
Γ
be
a
connected
semi-graph
[cf.
[SemiAn],
§1].
Then
we
shall
refer
to
the
dimension
over
Q
of
the
first
homology
module
of
Γ
[with
coefficients
in
Q]
H
1
(Γ,
Q)
as
the
loop-rank
of
Γ.
We
shall
say
that
Γ
is
untangled
if
every
closed
edge
of
Γ
abuts
to
two
distinct
vertices.
Schemes:
Let
K
be
a
field;
K
⊆
L
a
field
extension;
X
an
algebraic
variety
[i.e.,
a
separated,
geometrically
integral
scheme
of
finite
type]
over
K.
Then
we
shall
def
write
X
L
=
X
×
K
L.
Suppose
that
X
is
a
smooth
proper
curve
[i.e.,
a
smooth,
proper
algebraic
variety
of
dimension
1]
over
K.
Then
we
shall
write
J(X)
for
the
Jacobian
of
X.
Let
p
be
a
prime
number;
A
a
semi-abelian
variety
or
a
p-divisible
group
over
K.
Then
we
shall
write
T
p
A
for
the
p-adic
Tate
module
associated
to
[the
p-power
torsion
points
valued
in
some
fixed
algebraic
closure
of
K
of]
A.
Suppose
that
K
is
a
valuation
field.
Let
X
be
an
O
K
-scheme.
Then
we
shall
write
X
s
for
the
special
fiber
of
X
[i.e.,
the
fiber
of
X
over
the
closed
point
of
Spec
O
K
].
Let
S
1
,
S
2
be
schemes.
Then
we
shall
write
Isom(S
1
,
S
2
)
for
the
set
of
isomorphisms
of
schemes
between
S
1
and
S
2
.
Curves:
We
shall
use
the
term
“hyperbolic
curve”
[i.e.,
a
family
of
hyperbolic
curves
over
the
spectrum
of
a
field]
as
defined
in
[MT],
§0.
We
shall
use
the
term
“n-th
configuration
space”
as
defined
in
[MT],
Definition
2.1,
(i).
Log
schemes:
If
X
log
is
a
fine
log
scheme,
then
we
shall
write
•
X
for
the
underlying
scheme
of
X
log
;
•
M
X
for
the
étale
sheaf
of
monoids
on
X
that
defines
the
log
structure
of
X
log
;
×
,
which
we
shall
refer
to
as
the
characteristic
of
X
log
[cf.
•
M
X
=
M
X
/O
X
[MT],
Definition
5.1,
(i)].
def
Let
K
be
a
complete
discrete
valuation
field;
Y
a
hyperbolic
curve
over
K;
Y
a
def
compactified
semistable
model
of
Y
over
O
K
[cf.
Definition
2.1,
(ii)].
Write
S
=
Spec
O
K
;
S
log
for
the
log
scheme
determined
by
the
log
structure
associated
to
the
closed
point
of
S.
Then
it
follows
immediately
from
[Hur],
§3.7,
§3.8,
13
that
the
multiplicative
monoid
of
sections
of
O
Y
that
are
invertible
on
[the
open
subscheme
of
Y
determined
by]
Y
determines
a
natural
log
structure
on
Y.
Denote
the
resulting
log
scheme
by
Y
log
.
Then
one
verifies
immediately
that
the
natural
morphism
of
schemes
Y
→
S
extends
to
a
a
proper,
log
smooth
morphism
Y
log
→
S
log
of
fine
log
schemes.
Let
y
be
a
geometric
point
of
Y
s
.
Write
M
y
pf
for
the
perfection
of
the
stalk
of
the
characteristic
M
Y
at
y.
Then
one
verifies
immediately
that,
if
the
image
of
y
in
Y
s
is
a
smooth
point
that
is
not
a
cusp
(respectively,
a
cusp;
a
node),
then
∼
∼
∼
M
y
pf
→
Q
≥0
(respectively,
M
y
pf
→
Q
≥0
×
Q
≥0
;
M
y
pf
→
Q
≥0
×
Q
≥0
).
Fundamental
groups:
For
a
connected
noetherian
scheme
S,
we
shall
write
Π
S
for
the
étale
funda-
mental
group
of
S,
relative
to
a
suitable
choice
of
basepoint.
Let
Σ
⊆
Primes
be
a
nonempty
subset;
K
a
perfect
field;
X
an
algebraic
variety
over
K.
Then
we
shall
write
def
Δ
X
=
Π
X
K
;
(Σ)
def
Π
X
=
Π
X
/Ker(Δ
X
Δ
Σ
X
),
Σ
where
Δ
X
Δ
Σ
X
denotes
the
natural
surjection.
We
shall
refer
to
Δ
X
(respec-
(Σ)
tively,
Π
X
)
as
the
geometric
pro-Σ
fundamental
group
(respectively,
geometri-
cally
pro-Σ
fundamental
group)
of
X.
Let
p
be
a
prime
number,
Σ
⊆
Primes
a
nonempty
subset.
Suppose
that
K
is
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p.
Write
Ω
for
the
p-adic
completion
of
K.
Let
F
be
one
of
the
[topological]
fields
K,
K,
and
Ω;
X
a
hyperbolic
curve
over
F
.
If
F
=
Ω,
then
we
shall
write
Π
tp
X
for
the
Σ-tempered
fundamental
group
of
X,
i.e.,
in
the
terminology
of
[CbTpIII],
Definition
3.1,
(ii),
the
“Σ-tempered
fundamental
group”
of
the
smooth
log
curve
over
F
determined
by
X
[where
we
note
that
one
verifies
immediately
that
the
∧
field
“
K
”
of
loc.
cit.
may
be
taken
to
to
be
the
field
F
=
Ω
of
the
present
discussion,
and
that
the
discussion
of
loc.
cit.
may
be
applied
to
the
situation
of
the
present
discussion
even
if
the
X
of
the
present
discussion
does
not
descend
to
the
field
“
K”
of
loc.
cit.].
Here,
we
recall
that,
when
F
=
Ω,
it
follows
immediately
from
[André],
Proposition
4.3.1
[and
the
surrounding
discussion],
that
the
universal
topological
coverings
of
finite
coverings
corresponding
to
char-
acteristic
open
subgroups
of
Π
tp
X
of
finite
index
determine
a
countable
collection
tp
of
characteristic
open
subgroups
of
Π
tp
X
that
form
a
basis
of
the
topology
of
Π
X
,
tp
hence
determine
a
natural
topology
on
Out(Π
X
).
Thus,
if
F
=
K,
then
we
obtain
a
natural
continuous
outer
action
G
K
Out(Π
tp
X
Ω
)
−→
14
and
hence,
since
Π
tp
X
Ω
is
center-free
[cf.,
e.g.,
[CbTpIII],
Proposition
3.3,
(i),
(ii);
[MT],
Proposition
1.4],
a
topological
group
Π
tp
X
def
=
out
Π
tp
X
Ω
G
K
,
which
we
refer
to
as
the
geometrically
Σ-tempered
fundamental
group
of
X.
Moreover,
if
F
=
K,
then
Π
tp
X
is
equipped
with
a
natural
continuous
surjection
tp
Π
X
G
K
,
whose
kernel
[which
in
fact
may
be
naturally
identified
with
Π
tp
X
Ω
]
we
def
=
Π
tp
denote
by
Δ
Σ-tp
X
X
Ω
and
refer
to
as
the
geometric
Σ-tempered
fundamental
group
of
X.
If
F
=
K,
then,
after
possibly
replacing
K
by
a
finite
extension
of
K,
we
may
assume
that
X
descends
to
a
hyperbolic
curve
X
K
over
K
[so
that
X
may
be
naturally
identified
with
“(X
K
)
K
”],
and
we
shall
refer
to
Π
tp
X
def
=
Π
tp
X
Ω
as
the
Σ-tempered
fundamental
group
of
X.
Finally,
we
note
that,
in
the
case
where
F
=
K,
it
follows
immediately
from
[CanLift],
Proposition
2.3,
(ii)
[cf.
also
[CbTpIII],
Proposition
3.3,
(i)],
together
with
the
definition
of
the
Σ-
tp
tempered
fundamental
group,
that
the
pro-Σ
completion
of
Π
tp
X
=
Π
X
Ω
may
be
naturally
identified,
for
suitable
choices
of
basepoints,
with
the
geometric
pro-Σ
fundamental
group
Δ
Σ
X
of
X.
1
Local
construction
of
Artin-Schreier
extensions
in
the
special
fiber
Let
p
be
a
prime
number.
In
the
present
section,
we
perform
various
local
computations
concerning
iterates
of
the
p-th
power
morphism
of
the
multiplica-
tive
group
scheme
G
m
over
the
ring
of
integers
of
a
mixed
characteristic
discrete
valuation
field
of
residue
characteristic
p.
As
a
consequence,
by
restricting
such
morphisms
to
suitable
formal
neighborhoods,
we
conclude
that
smooth
curves
of
genus
≥
1
appear
in
the
special
fibers
of
suitable
models
of
the
domain
curves
of
such
morphisms
[cf.
Proposition
1.6;
Remark
1.6.2].
This
observation
will
be
applied
in
§2
to
prove
that
arbitrary
hyperbolic
curves
over
p-adic
local
fields
ad-
mit
resolution
of
nonsingularities.
The
contents
of
this
section
may
be
regarded
as
an
alternative
and
somewhat
more
detailed
discussion
of
[Lpg1],
Proposition
2.4,
that
is
phrased
entirely
in
the
language
of
schemes
and
formal
schemes
and
does
not
resort
to
the
use
of
Berkovich
spaces.
First,
we
begin
with
several
elementary
lemmas
concerning
the
p-adic
valu-
ations
of
the
coefficients
of
certain
polynomials
and
power
series
[cf.
Lemmas
1.1,
1.2,
1.3].
15
Lemma
1.1.
Let
K
be
a
mixed
characteristic
discrete
valuation
field
of
residue
characteristic
p.
Suppose
that
K
contains
a
primitive
p-th
root
of
unity
ζ
p
∈
K.
def
Write
π
=
1
−
ζ
p
∈
K;
def
f
(x)
=
π
−p
(1
+
πx)
p
−
1
∈
K[x],
where
x
denotes
an
indeterminate.
Then
it
holds
that
f
(x)
−
(x
p
−
x)
∈
m
K
[x]
⊆
O
K
[x].
def
Proof.
First,
write
c
=
p
+
(−π)
p−1
.
Then
since
π
=
1
−
ζ
p
∈
K,
it
holds
that
c
=
p
+
(−π)
p−1
=
p
+
(−π)
p−1
+
π
−1
((1
−
π)
p
−
1).
Observe
that
these
equalities
imply
that
v
p
(c)
>
1.
In
particular,
it
holds
that
v
p
(p
+
(−π)
p−1
)
>
1,
hence
that
v
p
(p
+
π
p−1
)
>
1.
Next,
observe
that,
if
we
write
f
(x)
=
a
i
x
i
∈
K[x],
1≤i≤p
then
since
v
p
(p)
=
1
=
v
p
(π
p−1
),
and
v
p
(p
+
π
p−1
)
>
1,
it
holds
that
a
1
=
p
,
π
p−1
a
p
=
1,
v
p
(a
1
+
1)
>
0,
v
p
(a
i
)
>
0
(∀i
=
1,
p).
Thus,
we
conclude
that
f
(x)
−
(x
p
−
x)
∈
m
K
[x].
This
completes
the
proof
of
Lemma
1.1.
Lemma
1.2.
Let
K
be
a
mixed
characteristic
discrete
valuation
field
of
residue
characteristic
p;
n
a
positive
integer.
Write
q
i
x
i
∈
K[[x]]
f
(x)
=
1
+
i≥1
—
where
x
denotes
an
indeterminate
—
for
the
n-th
root
of
1+x
∈
K[[x]]
whose
constant
term
is
equal
to
1.
Then
it
holds
that
v
p
(q
1
)
=
−v
p
(n),
v
p
(q
i
)
≥
−iv
p
(n)
−
v
p
(i!)
≥
−i
v
p
(n)
+
1
p
−
1
(∀i
≥
2)
i
[cf.
the
well-known
elementary
fact
that
v
p
(i!)
≤
p−1
].
If,
moreover,
n
is
prime
to
p,
then
v
p
(q
i
)
≥
0
for
each
positive
integer
i.
1
Proof.
First,
we
observe,
by
considering
the
Taylor
expansion
of
(1
+
x)
n
at
0,
that
1
1
−
k
q
i
=
i!
n
0≤k≤i−1
16
for
each
positive
integer
i.
The
equality
and
inequalities
of
the
second
display
of
the
statement
of
Lemma
1.2
follow
immediately.
Next,
suppose
that
n
is
prime
to
p.
For
each
positive
integer
m,
write
def
Q
m
⊆
K
for
the
O
K
-subalgebra
generated
by
{q
j
}
1≤j≤m−1
.
Write
Q
0
=
O
K
.
Then,
by
comparing
the
coefficients
of
x
m
in
the
left-
and
right-hand
sides
of
the
equality
n
1+
q
i
x
i
=
1
+
x,
i≥1
we
conclude
that
nq
m
∈
Q
m−1
.
In
particular,
since
n
is
prime
to
p,
it
holds
that
q
m
∈
Q
m−1
.
Thus,
by
induction,
we
conclude
that
q
i
∈
O
K
for
each
positive
integer
i.
This
completes
the
proof
of
Lemma
1.2.
Lemma
1.3.
Let
K
be
a
mixed
characteristic
discrete
valuation
field
of
residue
characteristic
p;
n
a
positive
integer;
a
i
s
i
∈
O
K
[[s]]
\
{1},
g(s)
=
1
+
i≥1
where
s
denotes
an
indeterminate.
Write
i
0
for
the
smallest
positive
integer
i
such
that
a
i
=
0;
0
s
for
the
O
K
-valued
point
of
Spec
O
K
[[s]]
obtained
by
mapping
s
→
0.
Then
the
following
hold:
(i)
Suppose
that
v
p
(a
i
)
≥
2v
p
(n)
for
each
i
≥
1.
Then
g(s)
admits
an
n-th
root
1+
b
i
s
i
∈
O
K
[[s]].
i≥1
1
(ii)
Suppose
that
there
exists
an
element
x
∈
O
K
such
that
v
p
(x)
=
3i
0
(p−1)
.
Then
there
exist
•
a
positive
integer
j,
2
≤
•
an
element
b
∈
O
K
satisfying
the
inequalities
v
p
(b)
≥
1
and
3(p−1)
1
v
p
(b)
−
v
p
(b)
<
p−1
,
and
∼
•
an
isomorphism
h
:
O
K
[[s]]
→
O
K
[[s]]
of
topological
O
K
-algebras
such
that
h
maps
0
s
→
0
s
and
h(g(x
j
s))
=
1
+
bs
i
0
.
(iii)
Suppose
that
g(s)
=
1+a
i
0
s
i
0
,
where
a
i
0
satisfies
the
inequalities
v
p
(a
i
0
)
≥
def
1
2
1
and
3(p−1)
≤
v
p
(a
i
0
)
−
v
p
(a
i
0
)
<
p−1
.
Write
μ
=
v
p
(a
i
0
)
−
1
≥
0.
μ
Then
g(s)
admits
a
p
-th
root
1
c
i
0
i
s
i
0
i
∈
O
K
[[s]],
g(s)
pμ
=
1
+
i≥1
p
2
2
≤
v
p
(c
i
0
)
<
p−1
,
v
p
(c
2i
0
)
≥
2
1
+
3(p−1)
−
1,
and
where
1
+
3(p−1)
1
v
p
(c
i
0
i
)
≥
i
1
−
3(p−1)
for
each
positive
integer
i
>
2.
17
(iv)
In
the
notation
of
(iii),
suppose
that
K
contains
a
primitive
p-th
root
of
unity
ζ
p
∈
K
and
an
element
c
∈
K
such
that
c
i
0
=
π
p
,
c
i
0
def
p
where
we
write
π
=
1
−
ζ
p
.
[Note
that
since
v
p
(c
i
0
)
<
p−1
,
it
holds
that
v
p
(c)
>
0.]
Write
1
d
i
0
i
s
i
0
i
=
π
−p
(g(cs)
pm
−
1)
∈
K[[s]].
def
i≥1
p
,
v
p
(d
2i
0
)
>
Then
it
holds
that
d
i
0
=
1,
v
p
(d
i
0
)
=
0
>
v
p
(c
i
0
)
−
p−1
p
2i
0
sup{v
p
(c
)
−
p−1
,
0},
and
v
p
(d
i
0
i
)
>
i
·
sup
1
−
1
p
,
v
p
(c
i
0
)
−
≥
0
3(p
−
1)
p
−
1
for
each
positive
integer
i
>
2.
Moreover,
v
p
(d
i
0
i
)
>
v
p
(c
i
0
i
)
for
each
sufficiently
large
positive
integer
i.
Proof.
First,
we
verify
assertion
(i).
Note
that
g(s)
admits
an
n-th
root
1+
b
i
s
i
∈
K[[s]].
i≥1
Thus,
it
suffices
to
verify
that
v
p
(b
i
)
≥
0
for
each
positive
integer
i.
Note
that
1
if
v
p
(n)
≥
1
(≥
p−1
),
then
2iv
p
(n)
−
i
v
p
(n)
+
1
p
−
1
≥
0.
Thus,
by
applying
Lemma
1.2
[where
we
take
“x”
to
be
the
element
i≥1
a
i
s
i
],
together
with
our
assumption
that
v
p
(a
i
)
≥
2v
p
(n)
for
each
positive
integer
i,
we
conclude
that
v
p
(b
i
)
≥
0
for
each
positive
integer
i.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Fix
an
element
x
∈
O
K
such
that
v
p
(x)
=
def
ij
1
3i
0
(p−1)
.
For
each
pair
of
positive
integers
(i,
j),
write
b
i,j
=
x
a
i
.
Note
that
it
follows
immediately
from
our
assumption
on
v
p
(x)
[i.e.,
by
thinking
of
the
real
1
line
R
modulo
integral
multiples
of
3(p−1)
=
i
0
v
p
(x)]
that
there
exists
a
positive
integer
j
such
that
•
v
p
(b
i
0
,j
)
≥
1,
2
1
•
3(p−1)
≤
v
p
(b
i
0
,j
)
−
v
p
(b
i
0
,j
)
<
p−1
,
and
18
•
v
p
(b
i,j
)
≥
v
p
(b
i
0
,j
)
+
2v
p
(i
0
)
for
each
positive
integer
i
>
i
0
.
def
Fix
such
a
positive
integer
j
and
write
b
=
b
i
0
,j
.
Then
the
existence
of
an
∼
isomorphism
h
:
O
K
[[s]]
→
O
K
[[s]]
of
topological
O
K
-algebras
such
that
h
maps
0
s
→
0
s
and
h(g(x
j
s))
=
1
+
bs
i
0
follows
immediately
from
Lemma
1.3,
(i),
where
we
take
“n”
to
be
i
0
and
“g(s)”
to
be
g
†
(s)
=
b
−1
s
−i
0
(g(x
j
s)
−
1)
def
1
def
[so
g(x
j
s)
=
1+bs
i
0
g
†
(s)].
[That
is
to
say,
h
is
defined
by
taking
h(s(g
†
(s))
n
)
=
1
s,
where
“(g
†
(s))
n
”
denotes
the
n-th
root
of
Lemma
1.3,
(i).]
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
Lemma
1.2
[applied
to
x
=
a
i
0
s
i
0
],
together
with
the
elementary
fact
that
v
p
(i!)
≤
1
for
i
=
2.
Finally,
we
verify
assertion
(iv).
It
follows
immediately
from
the
various
definitions
involved
that
p(i−1)
d
i
0
i
=
c
i
0
i
c
−i
.
i
0
π
1
In
particular,
it
holds
that
d
i
0
=
1.
Moreover,
since
v
p
(π)
=
p−1
,
and
v
p
(c
i
0
)
<
p
[cf.
Lemma
1.3,
(iii)],
it
holds
that
p−1
v
p
(d
i
0
i
)
=
v
p
(c
i
0
i
)
−
iv
p
(c
i
0
)
+
p(i
−
1)
p
>
v
p
(c
i
0
i
)
−
.
p
−
1
p
−
1
Suppose
that
i
=
2
(respectively,
i
≥
3).
Then
it
follows
immediately
from
Lemma
1.3,
(iii),
that
v
p
(c
i
0
i
)
−
2
p
≥
2
1+
p
−
1
3(p
−
1)
−
1
−
p
1
=
>
0
p
−
1
3(p
−
1)
(respectively,
v
p
(c
i
0
i
)−
1
1
2p
−
4
p
p
p
≥
i
1−
≥
3
1−
=
≥
0).
−
−
p
−
1
3(p
−
1)
p
−
1
3(p
−
1)
p
−
1
p
−
1
Thus,
we
conclude
that
v
p
(d
2i
0
)
>
0,
and
v
p
(d
i
0
i
)
>
i
1
−
1
3(p
−
1)
−
p
≥
0
p
−
1
for
each
positive
integer
i
>
2.
The
remainder
of
assertion
(iv)
follows
immedi-
ately
from
the
inequalities
already
obtained,
together
with
the
inequalities
p
1
1
p
>
inf
1
−
,
+
p
−
1
3(p
−
1)
2
·
3(p
−
1)
2(p
−
1)
p
2
1
=
−
1+
>
3(p
−
1)
p
−
1
3(p
−
1)
≥
v
p
(c
i
0
)
[cf.
Lemma
1.3,
(iii)].
This
completes
the
proof
of
assertion
(iv),
hence
of
Lemma
1.3.
19
Definition
1.4.
Let
K
be
a
mixed
characteristic
discrete
valuation
field
of
residue
characteristic
p;
c
∈
m
K
\
{0}.
Then
we
shall
write
φ
c
:
B
c
−→
Spec
O
K
[[t]]
—
where
t
denotes
an
indeterminate
—
for
the
blow-up
of
Spec
O
K
[[t]]
with
center
given
by
the
closed
subscheme
defined
by
the
ideal
(c,
t);
b
c
∈
B
c
for
the
generic
point
of
the
exceptional
irreducible
component
of
the
special
fiber
of
B
c
[i.e.,
the
fiber
of
B
c
over
the
closed
point
of
Spec
O
K
];
U
c
=
Spec
O
K
[[t]][s
c
]/(cs
c
−
t)
⊆
B
c
—
where
s
c
denotes
an
indeterminate,
but,
by
a
slight
abuse
of
notation,
we
shall
also
use
the
notation
“s
c
”
to
denote
the
element
of
Γ(U
c
,
O
U
c
)
determined
by
the
indeterminate
“s
c
”—
for
the
open
subscheme
obtained
by
removing
the
strict
transform
of
the
special
fiber
of
Spec
O
K
[[t]].
Note
that
it
follows
immediately
from
the
various
definitions
involved
that
φ
c
induces
a
morphism
c
def
=
Spec
O
K
[[s
c
]]
−→
Spec
O
K
[[t]]
φ
c
:
U
over
O
K
that
maps
t
to
cs
c
.
Proposition
1.5
(Local
construction
of
Artin-Schreier
extensions
in
the
special
fiber
I).
We
maintain
the
notation
of
Definition
1.4.
Suppose
that
K
contains
a
primitive
p-th
root
of
unity
ζ
p
∈
K.
Write
k
for
the
residue
def
field
of
K;
π
=
1
−
ζ
p
∈
K;
f
:
Spec
O
K
[[t]]
−→
Spec
O
K
[[t]]
for
the
[manifestly]
finite
flat
morphism
over
O
K
corresponding
to
the
homo-
morphism
of
topological
O
K
-algebras
that
maps
t
→
(1+t)
p
−1.
Then
f
induces
a
finite
morphism
f
˜
:
B
π
−→
B
π
p
over
O
K
that
maps
b
π
→
b
π
p
and
induces
a
finite
flat
morphism
U
π
→
U
π
p
,
whose
induced
morphism
on
special
fibers
is
the
morphism
induced
on
spectra
by
the
injective
homomorphism
k[s
π
p
]
−→
k[s
π
]
over
k
that
maps
s
π
p
→
s
pπ
−
s
π
.
Proof.
First,
observe
that
we
have
inclusions
of
ideals
in
O
K
[[t]]
2
(π,
t)
p
+p
⊆
(π
p
,
(1
+
t)
p
−
1)
⊆
(π,
t).
Indeed,
the
inclusion
(π
p
,
(1
+
t)
p
−
1)
⊆
(π,
t)
is
immediate.
Next,
we
verify
the
2
def
inclusion
(π,
t)
p
+p
⊆
(π
p
,
(1
+
t)
p
−
1).
Write
t̃
=
(1
+
t)
p
−
1.
Then
it
holds
20
2
2
2
that
t̃
p
∈
t
p
+
(p
2
)
⊆
t
p
+
(π
p
).
In
particular,
it
holds
that
t
p
∈
(
t̃
p
,
π
p
)
⊆
2
(π
p
,
t̃).
Thus,
we
conclude
that
π
i
t
p
+p−i
∈
(π
p
,
t̃)
for
each
integer
i
such
that
2
0
≤
i
≤
p
2
+
p,
hence
that
(π,
t)
p
+p
⊆
(π
p
,
(1
+
t)
p
−
1).
This
completes
the
proof
of
the
observation.
The
observation
of
the
preceding
paragraph,
together
with
Lemma
1.1,
im-
plies,
by
the
definition
of
the
blow-up,
that
the
morphism
f
functorially
induces
a
[proper
and
quasi-finite,
hence]
finite
morphism
f
˜
:
B
π
→
B
π
p
over
O
K
,
which,
in
turn,
induces
a
[manifestly]
finite
flat
morphism
U
π
−→
U
π
p
over
O
K
such
that
t
→
(1
+
t)
p
−
1,
s
π
p
→
π
−p
(1
+
πs
π
)
p
−
1
[cf.
Lemma
1.1].
Indeed,
it
follows
from
Lemma
1.1
that
this
morphism
U
π
→
U
π
p
determines
a
dominant
morphism
Spec
k[s
π
]
−→
Spec
k[s
π
p
]
[between
open
subschemes
of
the
respective
exceptional
irreducible
components
of
the
special
fibers
of
B
π
,
B
π
p
]
that
corresponds
to
the
injective
homomorphism
k[s
π
p
]
−→
k[s
π
]
over
k
that
maps
s
π
p
→
s
pπ
−
s
π
.
This
completes
the
proof
of
Proposition
1.5.
Remark
1.5.1.
In
the
notation
of
Proposition
1.5,
write
G
m
=
Spec
O
K
[u,
u
1
]
for
the
multiplicative
group
scheme
over
O
K
,
where
u
denotes
an
indeterminate;
ι
:
Spec
O
K
[[t]]
−→
G
m
for
the
morphism
that
corresponds
to
the
homomorphism
over
O
K
that
maps
u
→
1
+
t.
Then
we
have
a
commutative
diagram
U
π
⏐
⏐
f
˜
|
Uπ
−−−−→
Spec
O
K
[[t]]
−−−−→
G
m
ι
φ
π
|
U
π
⏐
⏐
⏐
⏐
p
f
U
π
p
−−−−−→
Spec
O
K
[[t]]
−−−−→
G
m
,
φ
π
p
|
U
π
p
ι
where
the
right-hand
vertical
arrow
denotes
the
p-th
power
morphism;
the
verti-
cal
arrows
are
finite
flat
morphisms
of
degree
p
[cf.
Proposition
1.5];
the
second
square
is
cartesian;
the
first
square
is
cartesian,
up
to
taking
the
normalization
of
the
fiber
product
that
would
make
the
first
square
“truly
cartesian”.
21
Proposition
1.6
(Local
construction
of
Artin-Schreier
extensions
in
the
special
fiber
II).
We
maintain
the
notation
of
Remark
1.5.1.
Let
a
i
t
i
∈
O
K
[[t]]
\
{1}.
g(t)
=
1
+
i≥1
Write
i
0
for
the
smallest
positive
integer
i
such
that
a
i
=
0;
λ
g
:
Spec
O
K
[[t]]
−→
Spec
O
K
[[t]]
for
the
morphism
over
O
K
corresponding
to
the
homomorphism
of
topological
O
K
-algebras
that
maps
t
→
g(t)
−
1.
Then
the
following
hold:
(i)
After
possibly
replacing
K
by
a
suitable
finite
field
extension
of
K,
there
exist
a
positive
integer
μ,
elements
c
1
,
c
2
∈
m
K
\
{0},
an
isomorphism
∼
λ
h
:
Spec
O
K
[[s
c
1
]]
→
Spec
O
K
[[s
c
1
]]
over
O
K
,
and
a
morphism
ξ
g
:
Spec
O
K
[[s
c
1
]]
−→
G
m
over
O
K
satisfying
the
following
conditions:
•
Write
0
c
1
for
the
O
K
-valued
point
of
Spec
O
K
[[s
c
1
]]
obtained
by
map-
ping
s
c
1
→
0.
Then
λ
h
maps
0
c
1
→
0
c
1
,
and
ξ
g
maps
0
c
1
to
the
identity
element
of
G
m
(O
K
).
•
There
exists
a
commutative
diagram
Spec
O
K
[[s
c
1
]]
−−−−→
Spec
O
K
[[t]]
−−−−→
Spec
O
K
[[t]]
c
φ
1
⏐
λ
h
⏐
λ
g
⏐
⏐
ι
Spec
O
K
[[s
c
1
]]
−−−−→
ξ
g
G
m
−−−
μ
−→
p
G
m
,
where
the
right-hand
lower
horizontal
arrow
denotes
the
p
μ
-th
power
morphism.
•
Write
∼
τ
:
Spec
O
K
[[t]]
→
Spec
O
K
[[s
c
1
]]
for
the
isomorphism
over
O
K
corresponding
to
the
isomorphism
of
topological
O
K
-algebras
that
maps
s
c
1
→
t;
η(s
c
2
)
∈
O
K
[[s
c
2
]]
for
the
image
of
u
via
the
homomorphism
O
K
[u,
u
1
]
→
O
K
[[s
c
2
]]
in-
duced
by
the
composite
∼
Spec
O
K
[[s
c
2
]]
−→
Spec
O
K
[[t]]
−→
Spec
O
K
[[s
c
1
]]
−→
G
m
.
c
φ
2
τ
22
ξ
g
Then
it
holds
that
η(s
c
2
)
−
1
∈
m
K
[[c
2
s
c
2
]],
and,
moreover,
π
−p
(η(s
c
2
)
−
1)
−
s
i
c
0
2
∈
m
K
[s
c
2
]
+
m
K
[[c
2
s
c
2
]]
=
m
K
[s
c
2
]
+
m
K
[[t]].
In
particular,
there
exists
a
morphism
θ
g
:
U
c
2
−→
U
π
p
over
O
K
that
fits
into
the
following
commutative
diagram
∼
U
c
2
−−−−−→
Spec
O
K
[[t]]
−−−−→
Spec
O
K
[[s
c
1
]]
τ
φ
c
2
|
U
c
2
⏐
⏐
⏐
⏐
θ
g
ξ
g
U
π
p
−−−−−→
Spec
O
K
[[t]]
−−−−→
φ
π
p
|
U
π
p
ι
G
m
.
(ii)
Fix
a
collection
of
data
(μ,
c
1
,
c
2
,
λ
h
,
ξ
g
)
as
in
(i).
Write
def
Y
=
U
c
2
×
U
π
p
U
π
for
the
fiber
product
determined
by
the
morphism
θ
g
:
U
c
2
→
U
π
p
and
the
morphism
U
π
→
U
π
p
induced
by
f
˜
[cf.
Proposition
1.5].
Then
the
natural
morphism
Y
s
−→
(U
c
2
)
s
=
Spec
k[s
c
2
]
induced
by
the
first
projection
morphism
Y
→
U
c
2
corresponds
to
the
nat-
ural
injective
homomorphism
k[s
c
2
]
→
k[s
c
2
,
y]/(y
p
−
y
−
s
i
c
0
2
)
over
k,
where
y
denotes
an
indeterminate.
Proof.
First,
we
consider
assertion
(i).
We
begin
by
applying
Lemma
1.3,
(ii),
where
we
take
“g(s)”
to
be
g(t)
[i.e.,
so
the
indeterminate
“s”
corresponds
to
t],
and
we
observe
that,
by
replacing
K
by
a
suitable
finite
extension
of
K,
we
may
assume
without
loss
of
generality
that
there
exists
an
“x”
as
in
Lemma
1.3,
∼
(ii).
This
yields
an
isomorphism
“h
:
O
K
[[s]]
→
O
K
[[s]]”
as
in
Lemma
1.3,
(ii),
whose
induced
morphism
on
spectra
—
where
we
interpret
the
indeterminate
“s”
to
be
s
c
1
,
and
we
take
“x
j
”
to
be
c
1
[so
“x
j
s”
corresponds
to
c
1
s
c
1
=
t]
—
we
take
to
be
λ
h
.
Here,
we
recall
that
this
isomorphism
“h”
of
Lemma
1.3,
(ii),
satisfies
a
condition
“h(g(x
j
s))
=
1
+
bs
i
0
”.
Next,
we
would
like
to
apply
Lemma
1.3,
(iii),
where
we
take
“1
+
a
i
0
s
i
0
”
to
be
the
“1
+
bs
i
0
”
of
Lemma
1.3,
(ii)
[i.e.,
so
the
indeterminate
“s”
still
corresponds
to
s
c
1
].
This
yields
a
1
power
series
“g(s)
pμ
”
as
in
Lemma
1.3,
(iii).
We
then
take
the
“μ”
of
Lemma
1.3,
(iii),
to
be
μ
and
define
ξ
g
to
be
the
morphism
over
O
K
corresponding
1
to
the
homomorphism
that
maps
u
to
this
power
series
“g(s)
pμ
”
[i.e.,
where
the
indeterminate
“s”
still
corresponds
to
s
c
1
].
This
yields
a
collection
of
data
23
(μ,
c
1
,
λ
h
,
ξ
g
)
that
satisfies
the
first
two
itemized
conditions
of
Proposition
1.6,
(i).
The
third
[and
final]
itemized
condition
of
Proposition
1.6,
(i),
now
follows
by
translating
the
various
estimates
of
Lemma
1.3,
(iv),
into
the
notation
of
the
present
situation,
where
we
take
the
“c”
of
Lemma
1.3,
(iv),
to
be
c
2
,
and
we
observe
that,
again
by
replacing
K
by
a
suitable
finite
extension
of
K,
we
may
assume
without
loss
of
generality
that
there
exist
“ζ
p
”
and
“c”
as
in
Lemma
1
1.3,
(iv).
Also,
we
observe
that
the
power
series
“g(cs)
pμ
”
of
Lemma
1.3,
(iv),
corresponds
to
η(s
c
2
)
[i.e.,
where
the
indeterminate
“s”
corresponds
to
s
c
2
].
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Recall
that
•
the
morphism
θ
g
:
U
c
2
→
U
π
p
corresponds
to
the
homomorphism
of
topo-
logical
O
K
-algebras
O
K
[[t]][s
π
p
]/(π
p
s
π
p
−
t)
−→
O
K
[[t]][s
c
2
]/(c
2
s
c
2
−
t)
that
maps
s
π
p
→
π
−p
(η(s
c
2
)
−
1),
t
→
η(s
c
2
)
−
1,
while
•
the
morphism
U
π
→
U
π
p
corresponds
to
the
homomorphism
of
topological
O
K
-algebras
O
K
[[t]][s
π
p
]/(π
p
s
π
p
−
t)
−→
O
K
[[t]][s
π
]/(πs
π
−
t)
that
maps
s
π
p
→
π
−p
((1
+
πs
π
)
p
−
1),
Then
since
and
t
→
(1
+
t)
p
−
1.
π
−p
((1
+
πs
π
)
p
−
1)
−
(s
pπ
−
s
π
)
∈
m
K
[[s
π
]]
π
−p
(η(s
c
2
)
−
1)
−
s
i
c
0
2
∈
m
K
[s
c
2
]
+
m
K
[[t]]
[cf.
Lemma
1.1;
Proposition
1.6,
(i)],
it
holds
that
•
the
morphism
(U
c
2
)
s
→
(U
π
p
)
s
corresponds
to
the
homomorphism
k[s
π
p
]
−→
k[s
c
2
]
over
k
that
maps
s
π
p
→
s
i
c
0
2
,
while
•
the
morphism
(U
π
)
s
→
(U
π
p
)
s
corresponds
to
the
homomorphism
k[s
π
p
]
−→
k[s
π
]
over
k
that
maps
s
π
p
→
s
pπ
−
s
π
.
24
Thus,
we
conclude
that
the
first
projection
morphism
Y
s
=
(U
c
2
)
s
×
(U
πp
)
s
(U
π
)
s
→
(U
c
2
)
s
=
Spec
k[s
c
2
]
corresponds
to
the
natural
injective
homomorphism
k[s
c
2
]
→
k[s
c
2
,
y]/(y
p
−
y
−
s
i
c
0
2
)
over
k.
This
completes
the
proof
of
assertion
(ii),
hence
of
Proposition
1.6.
Remark
1.6.1.
In
the
notation
of
Proposition
1.6,
we
observe
that
the
first
projection
morphism
Y
→
U
c
2
fits
into
a
commutative
diagram
Y
−−−−→
U
π
θ
Y
⏐
⏐
⏐
⏐
f
Y
f
˜
|
Uπ
−−−−→
Spec
O
K
[[t]]
−−−−→
G
m
ι
φ
π
|
U
π
⏐
⏐
⏐
⏐
p
f
U
c
2
−−−−→
U
π
p
−−−−−→
Spec
O
K
[[t]]
−−−−→
G
m
,
θ
g
φ
π
p
|
U
π
p
ι
where
the
first
vertical
arrow
f
Y
denotes
the
first
projection
morphism
Y
→
U
c
2
;
the
left-hand
upper
horizontal
arrow
θ
Y
denotes
the
second
projection
morphism
Y
→
U
π
;
the
vertical
arrows
are
finite
flat
morphisms
of
degree
p
[cf.
Proposition
1.5,
Remark
1.5.1];
the
first
and
third
squares
are
cartesian;
the
second
square
is
cartesian,
up
to
taking
the
normalization
of
the
fiber
product
that
would
make
the
second
square
“truly
cartesian”.
Remark
1.6.2.
Let
k
be
a
field
of
characteristic
p;
n
a
positive
integer.
Write
C
for
the
Artin-Schreier
curve
over
k
defined
by
the
equation
y
p
−
y
=
x
n
,
where
x
and
y
are
indeterminates;
g
C
for
the
genus
of
C.
Then
it
follows
immediately
from
Hurwitz’s
formula
that
g
C
=
(n
−
1)(p
−
1)
,
2
where
n
denotes
the
greatest
positive
integer
that
divides
n
and
is
prime
to
p.
In
particular,
if
n
is
not
a
power
of
p,
then
g
C
≥
1.
[Indeed,
by
considering
the
Frobenius
morphism,
one
reduces
immediately
to
the
case
where
n
=
n
.
Moreover,
the
computation
of
g
C
is
immediate
when
n
=
1
in
light
of
the
form
of
the
equation
y
p
−
y
=
x.
Thus,
the
computation
of
g
C
reduces
to
the
computation
of
the
genus
of
a
tamely
ramified
cyclic
covering
of
the
projective
line
of
degree
n
whose
ramification
consists
solely
of
p+1
totally
ramified
points.]
25
2
Resolution
of
nonsingularities
for
arbitrary
hy-
perbolic
curves
over
p-adic
local
fields
Let
p
be
a
prime
number.
In
the
present
section,
we
apply
certain
con-
structions
involving
p-divisible
groups
to
extend
the
Artin-Schreier
coverings
constructed
locally
in
§1
to
coverings
of
an
arbitrary
hyperbolic
curve.
As
a
consequence,
we
prove
that
arbitrary
hyperbolic
curves
over
p-adic
local
fields
satisfy
RNS,
i.e.,
“resolution
of
nonsingularities”
[cf.
Definition
2.2,
(vii);
The-
orem
2.17].
This
result
may
be
regarded
as
a
generalization
of
results
obtained
by
A.
Tamagawa
and
E.
Lepage
[cf.
[Tama2],
Theorem
0.2;
[Lpg1],
Theorem
2.7].
Historically
[cf.,
e.g.,
the
discussion
in
the
Introduction
to
[Tama2]],
the
roots
of
these
results
of
Tamagawa
and
Lepage
may
be
traced
back
to
the
tech-
nique
of
“passing
to
a
covering
with
singular
reduction
of
a
given
curve
with
smooth
reduction
over
a
p-adic
local
field”
applied
in
the
proof
of
[PrfGC],
The-
orem
9.2
[cf.
also
Proposition
2.3,
(xii),
below].
Moreover,
the
techniques
of
[AbsTopII],
§2,
may
be
regarded
as
a
sort
of
weak,
pro-p
version
of
Tamagawa’s
RNS
[cf.
[AbsTopII],
Remark
2.6.1].
In
fact,
the
approach
of
the
present
sec-
tion
may
be
regarded
as
a
sort
of
amalgamation
of
the
techniques
of
[Lpg1]
with
the
techniques
of
[AbsTopII],
§2.
At
any
rate,
from
a
historical
point
of
view,
it
is
interesting
to
observe
how
various
RNS
results
have
been
motivated
by
and
indeed
are
deeply
intertwined
with
various
results
in
anabelian
geometry
[cf.
Corollary
2.5,
as
well
as
Theorems
3.12,
3.13
in
§3
below].
First,
we
begin
by
fixing
our
conventions
concerning
models
of
hyperbolic
curves
[cf.
[DM],
Definition
1.1;
[Knud],
Definition
1.1].
Definition
2.1.
Let
K
be
a
valuation
field;
X
a
hyperbolic
curve
over
K;
X
a
scheme
over
O
K
.
Then:
(i)
We
shall
say
that
X
is
a
compactified
model
of
X
over
O
K
if
X
is
a
proper,
flat,
normal
scheme
of
finite
presentation
over
O
K
whose
generic
fiber
is
the
[uniquely
determined,
up
to
unique
isomorphism]
smooth
compactifi-
cation
of
X
over
K.
(ii)
Suppose
that
the
cusps
of
X
are
K-rational.
Then
we
shall
say
that
X
is
a
compactified
semistable
model
of
X
over
O
K
if
X
is
a
compactified
model
of
X
over
O
K
such
that
the
following
conditions
hold:
•
the
geometric
special
fiber
of
X
is
a
semistable
curve
[i.e.,
a
reduced,
connected
curve
each
of
whose
nonsmooth
points
is
an
ordinary
dou-
ble
point];
•
the
images
of
the
sections
Spec
O
K
→
X
determined
by
the
cusps
of
X
[which
we
shall
refer
to
as
cusps
of
X
]
lie
in
the
smooth
locus
of
X
and
do
not
intersect
each
other.
Suppose
that
X
is
a
compactified
semistable
model
of
X
over
O
K
.
Then
we
shall
say
that
X
has
split
reduction
if
X
s
is
split
[i.e.,
each
of
the
irreducible
components
and
nodes
of
X
s
is
geometrically
irreducible].
26
(iii)
Suppose
that
X
is
a
compactified
semistable
model
of
X
over
O
K
that
has
split
reduction.
Let
L
be
a
finite
extension
of
K
equipped
with
a
valuation
that
extends
the
valuation
on
K;
X
∗
a
compactified
semistable
model
of
X
L
over
O
L
such
that
X
∗
has
split
reduction
and
dominates
X
.
Then
we
shall
say
that
X
∗
is
a
toral
compactified
semistable
model
relative
to
X
if
each
irreducible
component
of
X
s
∗
that
maps
to
a
closed
point
of
X
s
[via
the
uniquely
determined
morphism
X
∗
→
X
]
is
normal
of
genus
0
and
has
precisely
2
nodes.
Suppose
that
X
∗
is
a
toral
compactified
semistable
model
relative
to
X
.
Let
v
be
a
vertex
of
the
dual
graph
associated
to
X
s
∗
.
Then
we
shall
say
that
v
is
a
toral
semistable
vertex
of
X
∗
if
v
corresponds
to
an
irreducible
component
of
X
s
∗
that
maps
to
a
closed
point
of
X
.
(iv)
We
shall
say
that
X
is
a
compactified
stable
model
of
X
over
O
K
if
X
is
a
compactified
semistable
model
of
X
over
O
K
such
that
X
,
together
with
the
cusps
of
X
,
determines
a
pointed
stable
curve.
(v)
We
shall
say
that
X
is
a
semistable
model
of
X
over
O
K
if
X
is
obtained
by
removing
the
cusps
from
a
[uniquely
determined,
by
Zariski’s
Main
Theorem,
up
to
unique
isomorphism]
compactified
semistable
model
of
X
over
O
K
.
Suppose
that
X
is
a
semistable
model
of
X
over
O
K
.
Then
we
shall
say
that
X
has
split
reduction
if
X
s
is
split
[i.e.,
each
of
the
irreducible
components
and
nodes
of
X
s
is
geometrically
irreducible].
(vi)
We
shall
say
that
X
is
a
stable
model
of
X
over
O
K
if
X
is
obtained
by
removing
the
cusps
from
a
[uniquely
determined,
by
Zariski’s
Main
Theorem,
up
to
unique
isomorphism]
compactified
stable
model
of
X
over
O
K
.
(vii)
We
shall
say
that
X
has
stable
reduction
over
K
if
there
exists
a
[necessarily
unique,
up
to
unique
isomorphism]
stable
model
of
X
over
O
K
.
Suppose
that
X
is
a
stable
model
of
X
over
O
K
.
Then
we
shall
say
that
X
has
split
stable
reduction
over
K
if
X
s
is
split
[i.e.,
each
of
the
irreducible
components
and
nodes
of
X
s
is
geometrically
irreducible].
Remark
2.1.1.
It
follows
from
elementary
commutative
algebra/scheme
theory
[cf.
[EGAIV
2
],
Corollaire
6.1.2;
[EGAIV
3
],
Proposition
12.1.1.5]
that
any
com-
pactified
model
as
in
Definition
2.1,
(i),
is
of
dimension
2
whenever
K
is
a
complete
discrete
valuation
field.
Remark
2.1.2.
In
the
notation
of
Definition
2.1,
suppose
that
X
is
a
proper
hyperbolic
curve
over
K.
Then
it
follows
immediately
from
the
various
defini-
tions
involved
that
the
notion
of
a
compactified
semistable
model
of
X
over
O
K
coincides
with
the
notion
of
a
semistable
model
of
X
over
O
K
.
27
Remark
2.1.3.
In
the
notation
of
Definition
2.1,
suppose
that
K
is
a
complete
discrete
valuation
field,
and
that
X
has
split
stable
reduction
over
K.
Let
X
be
a
compactified
semistable
model
of
X
over
O
K
that
has
split
reduction;
φ
:
Y
→
X
a
morphism
of
compactified
semistable
models
over
O
K
that
re-
stricts
to
a
connected
finite
étale
covering
Y
→
X
over
K.
Then
one
verifies
immediately
—
by
considering
the
map
induced
on
irreducible
components
of
the
respective
special
fibers
by
the
necessarily
finite,
hence
surjective
morphism
[induced
by
Y
→
X
]
between
the
[two-dimensional,
normal,
integral]
spectra
of
the
completions
of
the
local
rings
of
X
,
Y
at
the
closed
points
under
consider-
ation
—
that
φ
always
maps
a
smooth
closed
point
of
Y
that
is
isolated
in
the
fiber
of
φ
to
a
smooth
closed
point
of
X
.
Remark
2.1.4.
In
the
notation
of
Remark
2.1.3,
let
X
∗
be
a
toral
compactified
semistable
model
relative
to
X
;
e
an
edge
of
the
dual
graph
associated
to
X
s
;
b
a
branch
of
the
node
e;
v
a
toral
semistable
vertex
of
X
∗
that
maps
to
[i.e.,
for
which
the
corresponding
irreducible
component
maps
to
the
node
corresponding
e
of
the
local
ring
of
X
at
e
is
isomorphic
to
to]
e.
Recall
that
the
completion
O
O
K
[[x,
y]]/(xy
−
a),
where
a
∈
m
K
\
{0},
and
x,
y
denote
indeterminates
chosen
so
that
the
ideal
(x)
(⊆
O
K
[[x,
y]]/(xy
−
a))
corresponds
to
b.
Observe
that
this
ideal
(x)
is
independent
of
the
choice
of
x,
y
[cf.
[Hur],
§3.7,
Lemma].
In
particular,
we
obtain
a
homomorphism
of
local
rings
v
ψ
:
O
K
[[x,
y]]/(xy
−
a)
−→
O
v
denotes
the
completion
of
the
local
ring
of
X
∗
at
the
generic
point
where
O
of
the
irreducible
component
of
X
∗
corresponding
to
v.
Write
ord
v
(−)
for
the
v
whose
normalization
is
determined
by
normalized
valuation
associated
to
O
the
condition
that
ord
v
(p)
=
1
∈
R.
Thus,
we
obtain
a
rational
number
def
0
<
ρ
b,v
=
ord
v
(ψ(x))
<
1
ord
v
(ψ(a))
associated
to
b
and
v,
which
is
in
fact
independent
of
the
normalization
of
“ord
v
(−)”.
If,
moreover,
we
write
b
for
the
other
branch
of
e,
then
one
verifies
immediately
that
ρ
b,v
+
ρ
b
,v
=
1.
Finally,
we
observe
that
given
any
rational
number
ρ
such
that
0
<
ρ
<
1,
there
exist,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
an
X
∗
and
v
as
above
such
that
ρ
b,v
=
ρ.
Indeed,
it
follows
immediately
from
the
theory
of
pointed
stable
curves,
as
ex-
posed
in
[Knud],
that,
by
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K
and
X
by
a
suitable
dense
open
subscheme
of
X,
we
may
assume
that
X
is
the
[unique,
up
to
unique
isomorphism]
compactified
stable
model
of
X
over
O
K
,
and
that
ρ
may
be
written
as
a
fraction
whose
denominator
divides
the
positive
integer
v
p
(a).
Then
it
follows
again
from
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that,
if
we
take
28
•
X
∗
to
be
the
[unique,
up
to
unique
isomorphism]
compactified
stable
model
over
O
K
of
the
hyperbolic
curve
obtained
by
removing
from
X
a
suitable
K-rational
point
of
X
with
center
at
e
[cf.
the
construction
of
the
displayed
∼
r
X
†
,x
→)
homomorphism
“(
O
O
K
[[s,
t]]/(st
−
π
K
)
→
O
K
”
in
the
portion
of
the
proof
of
Proposition
2.3,
(iii),
below,
concerning
the
case
where
“x
is
a
nonsmooth
closed
point
of
X
†
”]
and
•
v
to
be
the
unique
irreducible
component
of
X
s
∗
that
maps
to
a
closed
point
of
X
s
via
the
natural
morphism
X
∗
→
X
[cf.
the
morphism
“X
†
[x]
→
X
†
”
in
the
portion
of
the
proof
of
Proposition
2.3,
(iii),
below,
concerning
the
case
where
“x
is
a
nonsmooth
closed
point
of
X
†
”],
then
ρ
b,v
=
ρ,
as
desired.
Remark
2.1.5.
In
the
notation
of
Remark
2.1.4,
we
observe
that,
for
a
fixed
choice
of
b,
the
assignment
v
→
ρ
b,v
∈
Q
that
assigns
to
a
toral
semistable
vertex
v
of
X
∗
that
maps
to
e
the
rational
number
ρ
b,v
is
injective.
Indeed,
it
follows
immediately
from
the
theory
of
pointed
stable
curves,
as
ex-
posed
in
[Knud]
—
i.e.,
by
adding
finitely
many
suitably
positioned
cusps
and
then
considering
the
various
contraction
morphisms
that
arise
from
eliminating
cusps
—
that,
to
verify
the
asserted
injectivity,
it
suffices
to
show
that
if
v
and
w
are
the
unique
toral
semistable
vertices
of
X
∗
that
map
to
e,
which
implies
that
there
exists
an
edge
e
∗
of
the
dual
graph
associated
to
X
s
∗
that
abuts
to
v
and
w,
then
ρ
b,v
=
ρ
b,w
.
To
this
end,
we
recall
that
X
and
X
∗
admit
natural
log
structures
[determined
by
the
respective
multiplicative
monoids
of
regular
functions
invertible
outside
the
respective
special
fibers
X
s
,
X
s
∗
]
such
that
the
morphism
X
∗
→
X
extends
uniquely
to
a
morphism
of
log
schemes
[cf.,
e.g.,
the
subsection
in
Notations
and
Conventions
entitled
“Log
schemes”;
the
discussion
e
∗
of
the
local
ring
of
of
[Hur],
§3.7,
§3.8,
§3.10].
Moreover,
the
completion
O
∗
∗
∗
∗
∗
∗
∗
X
at
e
is
isomorphic
to
O
K
[[x
,
y
]]/(x
y
−
a
),
where
a
∗
∈
O
K
,
and
x
∗
,
y
∗
denote
indeterminates
which
may
be
chosen
in
such
a
way
that
the
homo-
e
→
O
e
∗
induced
by
X
∗
→
X
maps
morphism
of
topological
O
K
-algebras
O
×
∗
N
∗
M
x
→
(x
)
·
(π
)
·
u,
for
some
unit
u
∈
O
e
∗
,
some
uniformizer
π
∗
∈
O
K
,
some
positive
integer
N
,
and
some
nonnegative
integer
M
.
[Indeed,
the
fact
that
N
is
necessarily
positive
follows
immediately,
in
light
of
our
assumptions
on
v
and
w,
from
well-known
considerations
in
intersection
theory
on
the
Q-factorial
normal
schemes
X
∗
and
X
.]
Then
the
desired
inequality
ρ
b,v
=
ρ
b,w
follows
immediately
from
the
fact
that,
up
to
a
possible
permutation
of
the
labels
“v”
and
“w”,
it
holds
that
x
∗
is
invertible
at
the
generic
point
of
[the
irreducible
component
corresponding
to]
v,
but
non-invertible
at
the
generic
point
of
[the
irreducible
component
corresponding
to]
w.
29
Remark
2.1.6.
In
the
notation
of
Remark
2.1.3,
suppose
further
that
Y
has
split
reduction,
and
that
the
morphism
Y
→
X
is
an
isomorphism.
Let
X
∗
be
a
toral
compactified
semistable
model
relative
to
X
,
Y
→
X
∗
a
morphism
over
X
.
Then
observe
that
it
follows
immediately
from
Remark
2.1.3,
together
with
Zariski’s
Main
Theorem,
that
each
normal
irreducible
component
of
Y
s
that
•
maps
to
a
closed
point
of
X
s
,
•
is
of
genus
0,
and
•
has
precisely
1
node
maps
to
a
closed
point
of
X
s
∗
.
In
particular,
it
follows
immediately
from
an
iterated
application
of
the
above
observation,
together
with
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud]
—
i.e.,
by
adding
finitely
many
suitably
positioned
cusps
and
then
considering
the
various
contraction
morphisms
that
arise
from
eliminating
cusps
—
that
there
exists
a
unique,
up
to
unique
isomorphism,
toral
compactified
semistable
model
Y
∗
relative
to
X
,
together
with
a
uniquely
deter-
mined
morphism
Y
→
Y
∗
of
compactified
semistable
models
over
X
,
such
that
the
following
universal
property
is
satisfied:
if
X
†
is
a
toral
compactified
semistable
model
relative
to
X
such
that
the
morphism
Y
→
X
admits
a
factorization
Y
→
X
†
→
X
,
then
the
morphism
Y
→
X
†
admits
a
unique
factorization
Y
→
Y
∗
→
X
†
.
That
is
to
say,
Y
∗
may
be
thought
of
as
a
sort
of
“universal
toralization
[over
X
]”
of
Y.
In
particular,
it
follows
immediately
from
the
existence
of
universal
toralizations,
together
with
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud]
—
i.e.,
by
adding
finitely
many
suitably
positioned
cusps
and
then
con-
sidering
the
various
contraction
morphisms
that
arise
from
eliminating
cusps
—
that
the
toral
compactified
semistable
models
relative
to
X
form
a
directed
inverse
system.
Definition
2.2.
Let
Σ
⊆
Primes
be
a
nonempty
subset;
K
a
mixed
character-
istic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
hyperbolic
curve
over
K.
Write
Ω
for
the
p-adic
completion
of
[some
fixed]
K.
Then:
(i)
Let
v
be
a
valuation
on
a
field
F
that
contains
K.
Write
O
v
for
the
ring
of
integers
determined
by
v;
m
v
⊆
O
v
for
the
maximal
ideal
of
O
v
.
Then
we
shall
say
that
v
is
a
p-valuation
[over
K]
if
O
K
=
O
v
∩
K
[which
implies
that
p
∈
m
v
].
Here,
the
phrase
“over
K”
will
be
omitted
in
situations
where
the
base
field
K
is
fixed
throughout
the
discussion.
We
shall
say
that
v
is
primitive
if
it
is
a
p-valuation
such
that
the
only
prime
ideal
of
O
v
that
contains
p
is
m
v
.
We
shall
say
that
v
is
residue-transcendental
def
if
it
is
a
p-valuation
whose
residue
field
k
v
=
O
v
/m
v
is
a
transcendental
extension
of
the
residue
field
of
K.
30
is
a
universal
(ii)
In
the
situation
of
(i),
suppose
that
v
is
a
p-valuation,
that
X
geometrically
pro-Σ
covering
of
X,
and
that
F
is
a
subfield
[in
a
fashion
Ω
)
compatible
with
the
given
inclusion
K
→
F
]
of
the
function
field
K(
X
of
X
Ω
.
Then
we
shall
say
that
v
is
point-theoretic
if
it
arises
from
some
point
x̃
∈
X(Ω),
i.e.,
if
O
v
⊆
F
is
equal
to
the
subring
of
F
consisting
Ω
that
are
regular
of
elements
∈
F
that
determine
rational
functions
on
X
at
x̃,
and
whose
value
at
x̃
is
contained
in
O
Ω
⊆
Ω.
Thus,
every
point
x̃
∈
X(Ω)
determines
a
corresponding
point-theoretic
valuation
of
F
.
(iii)
In
the
situation
of
(ii),
let
Z
→
X
be
a
connected
finite
étale
covering
→
Z
→
X.
Let
Z
be
a
compactified
equipped
with
a
factorization
X
semistable
model
of
Z
with
split
reduction.
Then
we
shall
write
VE(Z)
for
the
finite
set
[equipped
with
the
discrete
topology]
of
vertices
and
edges
of
the
dual
graph
associated
to
Z
s
.
Note
that
there
is
a
notion
of
specialization/generization
among
elements
of
VE(Z),
i.e.,
we
shall
say
that
•
a
vertex
specializes
to
a
node,
or,
alternatively,
that
a
node
generizes
to
a
vertex,
if
the
node
abuts
to
the
vertex;
•
a
vertex
specializes/generizes
to
a
vertex
if
the
two
vertices
coincide;
•
an
edge
specializes/generizes
to
an
edge
if
the
two
edges
coincide.
For
c
1
,
c
2
∈
VE(Z),
if
c
1
specializes
to
c
2
,
or,
equivalently,
c
2
generizes
to
c
1
,
then
we
shall
write
c
1
c
2
.
By
allowing
Z
and
Z
to
vary,
we
thus
obtain
a
topological
space
=
lim
VE(Z),
VE(
X)
←−
def
Z
where
the
transition
maps
in
the
inverse
limit
are
induced
by
the
corre-
sponding
scheme-theoretic
morphisms
of
compactified
semistable
models
[which
form
a
directed
inverse
system
—
cf.
Proposition
2.3,
(iii),
below],
that
is
to
say,
by
mapping
a
vertex
[i.e.,
irreducible
component]
or
edge
[i.e.,
node]
to
the
smallest
vertex
[i.e.,
irreducible
component]
or
edge
[i.e.,
node]
that
contains
its
scheme-theoretic
image.
[Here,
we
recall
that
any
such
morphism
of
compactified
semistable
models
always
maps
a
smooth
closed
point
that
is
isolated
in
the
fiber
of
the
morphism
to
a
smooth
closed
as
a
VE-
point
—
cf.
Remark
2.1.3.]
We
shall
refer
to
an
element
of
VE(
X)
chain
of
X.
Note
that
the
notion
of
specialization/generization
among
el-
ements
of
each
VE(Z)
determines
[i.e.,
by
considering
each
constituent
set
in
the
above
inverse
limit]
a
notion
of
specialization/generization
among
We
shall
say
that
an
element
c
∈
VE(
X)
is
primitive
elements
of
VE(
X).
if
every
generization
of
c
is
equal
to
c.
31
(iv)
In
the
situation
of
(iii),
let
z
c
1
,
z
c
2
∈
VE(Z).
Then
we
shall
write
δ(z
c
1
,
z
c
2
)
∈
Z
for
the
integer
δ
such
that
the
set
of
vertices
contained
in
a
path
of
minimal
length
between
z
c
1
and
z
c
2
on
the
dual
graph
of
Z
s
is
of
cardinality
δ
+
1.
Then
we
shall
write
Let
c
1
,
c
2
∈
VE(
X).
def
δ(c
1
,
c
2
)
=
sup
δ(w
c
1
,
w
c
2
)
∈
Z
∪
{+∞},
Z
where
Z
ranges
over
the
set
of
compactified
semistable
models
with
split
reduction
of
connected
finite
étale
coverings
Z
→
X
equipped
with
a
→
Z
→
X;
w
c
∈
VE(Z)
denotes
the
element
determined
factorization
X
i
by
c
i
for
each
i
=
1,
2.
(v)
In
the
situation
of
(ii),
suppose
further
that
F
contains
the
function
field
of
X.
Then
observe
that
v
determines,
by
considering
the
centers
K(
X)
associated
to
v
on
the
various
“Z”
in
the
discussion
of
(iii),
an
element
which
we
shall
refer
to
as
the
center-chain
associated
to
v.
∈
VE(
X),
In
particular,
any
point
x̃
∈
X(Ω)
determines,
by
considering
the
point-
which
we
shall
theoretic
valuation
associated
to
x̃,
an
element
∈
VE(
X),
refer
to
as
the
[point-theoretic]
center-chain
associated
to
x̃.
Write
x
∈
X(Ω)
for
the
image
of
x̃
in
X(Ω).
Thus,
the
Gal(
X/X
K
)-orbit
of
x̃
is
completely
determined
by
x.
We
shall
refer
to
the
Gal(
X/X
K
)-orbit
of
the
center-chain
associated
to
x̃
as
the
[point-theoretic]
orbit-center-chain
associated
to
x
[cf.
the
discussion
of
Remark
2.2.4
below].
(vi)
In
the
situation
of
(ii),
let
Z
→
X
be
a
connected
finite
étale
covering
→
Z
→
X.
Let
Z
be
a
compactified
equipped
with
a
factorization
X
semistable
model
with
split
reduction
of
Z.
In
the
remainder
of
the
dis-
cussion
of
the
present
item
(vi),
all
toral
compactified
semistable
models
relative
to
Z
will
be
assumed
to
have
generic
fibers
that
are
equipped
→
Z.
Write
with
the
structure
of
a
subcovering
of
the
pro-covering
X
V(Z)
(respectively,
E(Z))
for
the
set
of
vertices
(respectively,
edges)
of
Z
s
.
Thus,
VE(Z)
=
V(Z)
E(Z).
For
each
c
∈
VE(Z),
write
V
c
for
the
set
of
equivalence
classes
of
the
set
of
vertices
of
toral
compactified
semistable
models
relative
to
Z
that
map
to
the
closed
subscheme
of
Z
s
corresponding
to
c,
where
we
apply
the
equivalence
relation
induced
by
the
dominant
morphisms
over
Z
of
toral
compactified
semistable
models
relative
to
Z.
Let
e
∈
E(Z).
Then
we
shall
write
V
e
for
the
union
of
V
e
and
the
vertices
of
V(Z)
that
abut
to
e.
Observe
that,
for
each
toral
compactified
semistable
model
Z
†
relative
to
Z
and
32
each
c
†
∈
VE(Z
†
)
that
maps
to
the
closed
subscheme
of
Z
s
corresponding
to
c,
the
set
V
c
†
may
be
regarded,
in
a
natural
way
[i.e.,
by
considering
the
maps
induced
by
dominant
morphisms
over
Z
of
toral
compactified
semistable
models
relative
to
Z],
as
a
subset
of
V
c
.
We
shall
refer
to
such
a
subset
V
c
†
⊆
V
c
as
a
basic
open
subset
of
V
c
.
Thus,
from
the
point
of
view
of
the
natural
bijection,
determined
by
selecting
a
branch
b
of
the
edge
e,
between
V
e
and
the
set
of
rational
numbers
ρ
such
that
0
<
ρ
<
1
[cf.
Remarks
2.1.4,
2.1.5,
2.1.6],
the
basic
open
subsets
⊆
V
e
correspond
precisely
to
the
open
intervals
with
rational
endpoints
of
V
e
.
In
particular,
it
is
natural
to
regard
V
c
as
being
equipped
with
the
topology
determined
by
the
open
basis
consisting
of
the
basic
open
subsets
⊆
V
c
.
We
shall
refer
to
as
a
quasi-basic
open
subset
of
V
e
any
open
subset
of
V
e
which
is
a
union
of
a
countable
collection
of
basic
open
subsets
⊆
V
e
for
which
the
relation
of
inclusion
determines
a
total
ordering.
We
shall
refer
to
as
a
Dedekind
cut
of
V
e
an
unordered
pair
{D
1
,
D
2
}
of
disjoint
nonempty
quasi-basic
open
subsets
D
1
,
D
2
⊆
V
e
such
that
V
e
=
D
1
∪
D
2
.
Write
D
e
for
the
set
of
Dedekind
cuts
of
V
e
.
Note
that
the
topology
of
the
V
w
,
where
w
∈
V
e
\
V
e
,
induces,
in
a
natural
way,
a
topology
on
the
set
def
T
e
=
V
e
D
e
[i.e.,
by
taking
as
an
open
basis
for
the
topology
for
T
e
the
subsets
of
T
e
obtained
as
the
intersections
with
T
e
of
unions
of
an
open
subset
U
⊆
V
w
,
where
w
∈
V
e
\
V
e
,
with
the
set
of
Dedekind
cuts
{D
1
,
D
2
}
∈
D
e
such
that
both
D
1
and
D
2
intersect
U
].
Thus,
def
T
e
=
V
e
D
e
.
[with
the
induced
topology]
is
homeomorphic
to
the
open
interval
(0,
1)
⊆
R
of
the
real
line
[cf.
Remarks
2.1.4,
2.1.5,
2.1.6].
Write
def
VE(Z)
tor
=
V(Z)
∪
T
e
.
e∈E(Z)
Thus,
the
discrete
topology
on
V(Z),
together
with
the
topologies
defined
above
on
the
T
e
,
determine
a
topology
on
VE(Z)
tor
.
Moreover,
there
exists
a
noncontinuous
[cf.
Remark
2.2.1
below]
natural
surjective
map
Z
:
VE(Z)
tor
−→
VE(Z)
that
maps
each
T
e
to
e.
Finally,
by
allowing
Z
and
Z
to
vary,
we
thus
obtain
a
topological
space
tor
=
lim
VE(Z)
tor
VE(
X)
←−
def
Z
[cf.
the
discussion
of
(iii)],
together
with
a
natural
[not
necessarily
surjec-
tive!]
map
tor
−→
VE(
X).
X
:
VE(
X)
33
(vii)
We
shall
say
that
X
satisfies
Σ-RNS
[i.e.,
“Σ-resolution
of
nonsingulari-
ties”
—
cf.
[Lpg1],
Definition
2.1]
if
the
following
condition
holds:
Let
v
be
a
discrete
residue-transcendental
p-valuation
on
the
function
field
K(X)
of
X.
Then
there
exists
a
connected
geo-
metrically
pro-Σ
finite
étale
Galois
covering
Y
→
X
such
that
Y
has
stable
reduction
[over
its
base
field],
and
v
coincides
with
the
restriction
[to
K(X)]
of
a
discrete
valuation
on
the
function
field
K(Y
)
of
Y
that
arises
from
an
irreducible
component
of
the
special
fiber
of
the
stable
model
of
Y
.
Remark
2.2.1.
In
the
notation
of
Definition
2.2,
(vi),
the
natural
surjective
map
Z
is
not
continuous
in
general.
Indeed,
to
see
this,
it
suffices
to
observe
that
the
inverse
image
of
the
closed
subset
consisting
of
a
single
edge
is
an
open
subset
of
VE
tor
(Z)
that
is
not
closed.
Finally,
we
observe
that
one
may
also
conclude
from
this
noncontinuity
of
Z
that
X
is
not
continuous.
Remark
2.2.2.
In
the
notation
of
Definition
2.2,
(vii),
suppose
that
Σ
\
{p}
is
nonempty,
and
that
X
satisfies
Σ-RNS.
Then,
by
considering
a
suitable
admis-
sible
covering
of
the
stable
model
of
“Y
”
as
in
Definition
2.2,
(vii),
one
verifies
immediately
that
one
may
assume
that
the
normalization
of
the
irreducible
component
that
appears
in
Definition
2.2,
(vii),
is
of
genus
≥
2.
Remark
2.2.3.
In
the
notation
of
Definition
2.2,
(vii),
we
make
the
following
observations.
(i)
Let
L
⊆
Ω
be
a
topological
subfield
containing
K
that
arises
as
the
[topo-
logical]
field
of
fractions
of
a
mixed
characteristic
complete
discrete
valu-
ation
field
of
residue
characteristic
p.
Then
let
us
observe
that
any
compactified
semistable
model
of
X
L
over
O
L
arises,
after
possibly
replacing
K
and
L,
respectively,
by
suitable
finite
ex-
tension
fields
of
K
and
L,
as
the
result
of
base-changing,
from
O
K
to
O
L
,
some
compactified
semistable
model
of
X
K
over
O
K
.
Indeed,
since
every
element
of
L
admits
arbitrarily
close
p-adic
approxi-
mations
by
elements
of
finite
extension
fields
of
K
contained
in
K,
this
observation
follows
immediately
by
noting
that
it
follows
immediately
from
the
well-known
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that,
after
possibly
replacing
K
and
L,
respectively,
by
suitable
finite
extension
fields
of
K
and
L
and
possibly
replacing
X
by
some
dense
open
subscheme
of
X,
we
may
assume
without
loss
of
generality
that
the
given
compactified
semistable
model
of
X
L
over
O
L
is
in
fact
the
[unique,
up
to
unique
iso-
morphism]
compactified
stable
model
of
X
L
,
i.e.,
which
necessarily
arises
as
the
result
of
base-changing,
from
O
K
to
O
L
,
the
[unique,
up
to
unique
isomorphism]
compactified
stable
model
of
X
K
over
O
K
.
34
(ii)
We
maintain
the
notation
of
(i).
Then
let
us
observe
that
X
satisfies
Σ-RNS
if
and
only
if
X
L
satisfies
Σ-RNS.
Indeed,
this
observation
follows
immediately,
in
light
of
the
observation
of
(i),
from
Proposition
2.3,
(ii),
(iii),
below;
Proposition
2.4,
(iv),
below
[cf.
also
Definition
2.2,
(vii),
as
well
as
the
discussion
of
the
final
portion
of
the
subsection
in
Notations
and
Conventions
entitled
“Fundamental
groups”].
(iii)
Next,
let
L
be
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p
that
contains
K
as
a
topological
subfield.
Then
observe
that
it
follows
immediately
from
the
well-known
elementary
theory
of
complete
discrete
valuation
fields
that
L
is
isomorphic,
as
a
topological
K-algebra,
to
a
field
“L”
of
the
sort
discussed
in
(i),
(ii)
if
and
only
if
[the
valuation
on]
L
is
not
residue-transcendental
[relative
to
K],
i.e.,
if
and
only
if
the
residue
field
of
L
is
an
algebraic
extension
of
the
residue
field
of
K.
(iv)
We
maintain
the
notation
of
(iii).
Then
let
us
observe
that
if
X
L
satisfies
Σ-RNS,
then
the
residue
field
of
L
is
an
algebraic
extension
of
the
residue
field
of
K.
Indeed,
it
suffices
to
verify
this
observation
after
replacing
K
by
a
finite
extension
field
of
K.
In
particular,
we
may
assume
without
loss
of
gener-
ality
[cf.
Proposition
2.3,
(iii)]
that
there
exists
a
compactified
semistable
model
X
of
X
over
O
K
.
Then
it
follows
immediately
from
the
unique-
ness,
up
to
unique
isomorphism,
of
compactified
stable
models
[cf.
also
Definition
2.2,
(vii),
as
well
as
the
discussion
of
the
final
portion
of
the
subsection
in
Notations
and
Conventions
entitled
“Fundamental
groups”;
the
stable
reduction
theorem
of
[DM],
[Knud]],
that
if
X
L
satisfies
Σ-RNS,
then
any
closed
point
of
(X
⊗
O
K
O
L
)
s
that
arises
as
the
center
of
a
dis-
crete
residue-transcendental
p-valuation
on
the
function
field
of
X
L
is
necessarily
defined
over
some
algebraic
extension
of
the
residue
field
of
K.
On
the
other
hand,
this
contradicts
the
existence
of
discrete
residue-
transcendental
p-valuations
on
the
function
field
of
X
L
that
arise
as
the
local
rings
of
generic
points
of
exceptional
divisors
of
blow-ups
of
smooth
closed
points
of
(X
⊗
O
K
O
L
)
s
that
are
not
defined
over
some
algebraic
extension
of
the
residue
field
of
K.
This
completes
the
proof
of
the
above
observation.
(v)
We
maintain
the
notation
of
(iii).
Then
we
observe
further
that,
under
the
assumption
that
X
satisfies
Σ-RNS,
it
holds
that
the
residue
field
of
L
is
an
algebraic
extension
of
the
residue
field
of
K
if
and
only
if
X
L
satisfies
Σ-RNS.
35
Indeed,
necessity
follows
formally
from
the
observations
of
(ii),
(iii),
while
sufficiency
follows
formally
from
the
observation
of
(iv).
Remark
2.2.4.
In
the
context
of
Definition
2.2,
we
recall
from
the
general
theory
of
valuations
the
following
well-known
basic
facts.
Let
L
be
a
field
equipped
with
a
valuation
v,
M
a
finite
normal
extension
field
of
L.
Write
O
v
for
the
ring
of
integers
of
L
with
respect
to
v,
m
v
⊆
O
v
for
the
maximal
ideal
of
O
v
,
O
M
for
the
integral
closure
of
O
v
in
M
,
v
M/L
for
the
set
of
valuations
on
M
that
extend
v,
and
Aut(M/L)
for
the
group
of
automorphisms
of
M
that
restrict
to
the
identity
on
L.
If
w
∈
v
M/L
,
then
we
shall
write
O
w
for
the
ring
of
integers
of
def
M
with
respect
to
w,
m
w
⊆
O
w
for
the
maximal
ideal
of
O
w
,
p
w
=
O
M
∩
m
w
.
Then
the
set
v
M/L
is
nonempty
[cf.
[EP],
Theorem
3.1.1],
and
the
natural
action
of
Aut(M/L)
on
v
M/L
is
transitive
[cf.
[EP],
Theorem
3.2.14].
Moreover,
O
M
=
O
w
w∈v
M/L
[cf.
[EP],
Theorem
3.1.3,
(2)];
the
assignment
v
M/L
w
→
p
w
determines
a
bijective
correspondence
between
v
M/L
and
the
set
of
prime
ideals
of
O
M
that
lie
over
m
v
,
and,
for
w
∈
v
M/L
,
O
w
=
(O
M
)
p
w
[cf.
[EP],
Theorem
3.1.1;
[EP],
Theorem
3.2.13].
In
this
situation,
if
we
assume
further
that
v
is
real,
and
that
L
is
complete
with
respect
to
v,
then
v
M/L
is
of
cardinality
1
[cf.
[Neu],
Chapter
II,
Theorem
4.8].
[Here,
we
recall
that
if
v
is
not
real,
then
O
v
does
not,
in
general,
satisfy
Hensel’s
Lemma,
i.e.,
even
if
L
is
complete
with
respect
to
v
[cf.
[EP],
Remark
2.4.6].]
More
generally,
if
v
is
real,
then
L
admits
a
[cf.
[EP],
Theorem
1.1.4],
which
is
a
henselian
field
[cf.
natural
completion
L
[Neu],
Chapter
II,
Theorem
4.8;
the
discussion
preceding
[EP],
Lemma
4.1.1]
and
contains,
up
to
natural
isomorphism,
the
henselization
L
h
of
L
[cf.
[EP],
the
discussion
preceding
Theorem
5.2.2]
as
a
subfield,
i.e.,
L
h
⊆
L
[cf.
[EP],
Corollary
4.1.5;
[EP],
Corollary
5.2.3;
the
discussion
of
Case
2
in
the
proof
of
[EP],
Theorem
6.3.1].
The
various
basic
properties
stated
in
the
following
Proposition
2.3
consist
of
elementary
results
that
are
essentially
well-known
or
implicit
in
the
literature
[cf.
Remarks
2.3.2,
2.3.3
below],
but
we
give
[essentially]
self-contained
state-
ments
and
proofs
here
in
the
language
of
the
present
discussion
for
the
sake
of
completeness.
Proposition
2.3
(Basic
properties
of
models
of
hyperbolic
curves).
Let
K
be
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
charac-
teristic
p;
X
a
hyperbolic
curve
over
K.
Write
K(X)
for
the
function
field
of
X.
Then
the
following
hold:
36
(i)
Let
R
⊆
K(X)
be
a
finitely
generated
normal
O
K
-subalgebra
whose
field
of
fractions
coincides
with
K(X).
Then
Spec
R
arises
as
an
open
subscheme
of
a
compactified
model
of
X
over
O
K
.
(ii)
Let
v
be
a
discrete
residue-transcendental
[cf.
Remark
2.3.1
below]
p-
valuation
on
K(X).
Then
v
arises
as
the
discrete
valuation
associated
to
an
irreducible
component
of
the
special
fiber
of
a
compactified
model
of
X
over
O
K
.
(iii)
Let
X
be
a
compactified
model
of
X
over
O
K
equipped
with
the
action
of
a
finite
group
G
by
O
K
-linear
automorphisms
[which
thus
restrict
to
K-
linear
automorphisms
of
X].
Then,
after
possibly
replacing
K
by
a
finite
field
extension
of
K,
there
exists
a
compactified
semistable
model
of
X
over
O
K
that
dominates
X
and
is
stabilized
by
the
action
of
G
on
X.
(iv)
Let
Y
→
X
be
a
[connected]
finite
étale
Galois
covering
of
hyperbolic
curves
over
K,
Y
sst
a
compactified
semistable
model
of
Y
over
O
K
that
def
is
stabilized
by
G
=
Gal(Y
/X).
Write
X
for
the
quotient
of
Y
sst
by
the
natural
action
of
G
on
Y
sst
.
Then
X
is
a
compactified
semistable
model
of
X
over
O
K
,
and
the
images
of
smooth
points
of
Y
s
sst
via
the
natural
morphism
Y
s
sst
→
X
s
are
smooth
points
of
X
s
.
Moreover,
the
image
of
a
node
of
Y
s
sst
via
the
natural
morphism
Y
s
sst
→
X
s
is
a
node
of
X
s
if
and
only
if
G
does
not
permute
the
branches
of
the
node.
In
particular,
•
the
dual
graph
of
X
s
may
be
reconstructed
from
the
dual
graph
of
Y
s
sst
,
together
with
the
action
of
G
on
the
dual
graph
of
Y
s
sst
.
Finally,
•
this
reconstruction
procedure
is
functorial,
with
respect
to
maps
of
vertices/edges
to
vertices/edges
[i.e.,
as
in
the
discussion
of
Defi-
nition
2.2,
(iii)],
on
the
category
of
[connected]
finite
étale
Galois
coverings
of
X
over
K.
(v)
Suppose
that
we
are
in
the
situation
of
Definition
2.2,
(ii),
(iii),
(iv),
(v).
to
its
associated
Then
the
assignment
that
maps
a
p-valuation
on
K(
X)
center-chain
determines
bijections
as
follows:
p-valuations
on
K(
X)
∼
→
VE(
X),
primitive
p-valuations
on
K(
X)
∼
prim
,
→
VE(
X)
∼
pt-th
,
→
VE(
X)
X(Ω)
∼
pt-th
/Gal(
X/X
X(Ω)
→
VE(
X)
K
),
prim
⊆
VE(
X)
denotes
the
subset
of
primitive
VE-chains,
where
VE(
X)
pt-th
and
VE(
X)
⊆
VE(
X)
denotes
the
subset
of
point-theoretic
center-
chains.
37
Write
R
c
⊆
(vi)
Suppose
that
we
are
in
the
situation
of
(v).
Let
c
∈
VE(
X).
K(
X)
for
the
valuation
ring
of
the
p-valuation
associated
to
c
[cf.
(v)].
Then
it
holds
that
R
c
=
lim
O
Z,z
c
,
−→
Z
where
the
direct
limit
ranges
over
the
set
of
compactified
semistable
models
with
split
reduction
Z
of
the
domain
curves
of
connected
finite
étale
cov-
→
Z
→
X;
z
c
denotes
the
erings
Z
→
X
equipped
with
a
factorization
X
center
on
Z
determined
by
R
c
;
the
transition
maps
in
the
direct
limit
are
induced
by
the
corresponding
scheme-theoretic
morphisms
of
compactified
semistable
models
[cf.
the
discussion
of
Definition
2.2,
(iii)].
(vii)
Suppose
that
we
are
in
the
situation
of
(v).
Suppose,
moreover,
that
X
is
is
primitive
if
and
only
if
it
is
either
proper.
Then
a
p-valuation
of
K(
X)
real
or
point-theoretic.
Equivalently,
consists
of
the
disjoint
•
the
set
of
primitive
p-valuations
of
K(
X)
and
the
union
of
the
non-point-theoretic
real
p-valuations
of
K(
X)
point-theoretic
p-valuations
of
K(
X).
an
for
the
topological
pro-Berkovich
space
as-
In
particular,
if
we
write
X
sociated
to
[i.e.,
the
inverse
limit
of
the
underlying
topological
spaces
of
then
the
Berkovich
spaces
associated
to
the
finite
subcoverings
of
]
X,
may
be
naturally
identified
•
the
set
of
primitive
p-valuations
of
K(
X)
an
with
the
underlying
set
of
X
.
(viii)
Suppose
that
we
are
in
the
situation
of
(vii).
Then
there
exists
a
natural
commutative
diagram
of
maps
of
sets
an
X
⏐
⏐
∼
tor
−−−−→
VE(
X)
θ
X
⏐
⏐
X
prim
−−−−→
VE(
X),
VE(
X)
ι
X
where
the
upper
horizontal
arrow
θ
X
is
a
homeomorphism
[cf.
Remark
2.3.3
below];
the
lower
horizontal
arrow
ι
X
denotes
the
natural
inclu-
sion;
the
left-hand
vertical
arrow
denotes
the
bijection
obtained
by
form-
ing
the
composite
of
the
natural
identification
that
appears
in
the
state-
ment
of
(vi)
with
the
second
bijection
in
the
display
of
(v);
the
right-
hand
vertical
arrow
X
denotes
the
natural
morphism
[cf.
Definition
2.2,
(vi)].
In
particular,
X
is
injective
and
in
fact
admits
a
natural
split-
→
VE(
X)
tor
[i.e.,
such
that
τ
◦
is
the
identity
on
ting
τ
X
:
VE(
X)
X
X
tor
].
On
the
other
hand,
neither
ι
nor
is
surjective
[cf.
Remark
VE(
X)
X
X
2.3.4
below].
38
be
distinct
(ix)
Suppose
that
we
are
in
the
situation
of
(v).
Let
c
1
,
c
2
∈
VE(
X)
elements.
Then
one
of
the
following
conditions
holds:
•
δ(c
1
,
c
2
)
=
+∞.
such
•
δ(c
1
,
c
2
)
=
0,
and
there
exists
a
unique
element
c
3
∈
VE(
X)
that
c
3
c
1
and
c
3
c
2
.
In
particular,
if
c
1
and
c
2
are
distinct
primitive
elements,
then
it
holds
that
δ(c
1
,
c
2
)
=
+∞.
Then
the
(x)
Suppose
that
we
are
in
the
situation
of
(v).
Let
c
∈
VE(
X).
cardinality
of
the
set
\
{c}
|
c
c}
{c
∈
VE(
X)
is
at
most
1.
(xi)
Suppose
that
we
are
in
the
situation
of
(v).
Let
Σ
⊆
Primes
be
a
subset;
l
∈
Σ
\
{p};
H
⊆
G
K
a
closed
subgroup
such
that
the
restriction
to
H
of
the
l-adic
cyclotomic
character
of
K
has
open
image,
and,
moreover,
the
intersection
H
∩
I
K
of
H
with
the
inertia
subgroup
I
K
of
G
K
admits
(Σ)
a
surjection
to
[the
profinite
group]
Z
l
;
s
:
H
→
Π
X
a
section
of
the
(Σ)
restriction
to
H
of
the
natural
surjection
Π
X
G
K
.
Then
there
exists
prim
that
is
fixed
by
the
restriction,
via
s,
to
H
of
an
element
c
∈
VE(
X)
(Σ)
prim
⊆
VE(
X).
In
particular,
if
X
is
the
natural
action
of
Π
X
on
VE(
X)
an
an
proper,
then
there
exists
an
element
c
∈
X
[cf.
(vii)]
that
is
fixed
by
(Σ)
the
restriction,
via
s,
to
H
of
the
natural
action
of
Π
X
on
the
topological
an
.
pro-Berkovich
space
X
(xii)
Let
Σ
⊆
Primes
be
a
subset
of
cardinality
≥
2
that
contains
p.
Then
there
exists
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
X
†
→
X
satisfying
the
following
conditions:
•
X
†
has
split
stable
reduction.
•
Write
X
†
for
the
[unique,
up
to
unique
isomorphism]
stable
model
of
X
†
.
Then
X
s
†
is
singular,
and
every
irreducible
component
of
X
s
†
is
a
smooth
curve
of
genus
≥
2.
Proof.
First,
we
verify
assertion
(i).
Since
R
is
a
finitely
generated
algebra
over
O
K
(⊆
R),
it
follows
that
Spec
R
admits
an
embedding
over
O
K
into
†
N
N
-dimensional
affine
space
A
N
O
K
for
some
positive
integer
N
.
Write
Z
⊆
P
O
K
N
N
for
the
scheme-theoretic
closure
of
the
image
of
Spec
R
in
P
O
K
(⊇
A
O
K
);
Z
for
the
normalization
of
Z
†
.
Thus,
the
structure
sheaf
O
Z
is
p-torsion-free,
hence
flat
over
O
K
.
Since,
moreover,
Z
†
is
[of
finite
type
over
the
complete
discrete
valuation
ring
O
K
,
hence]
excellent,
it
follows
that
Z
is
a
proper,
flat
scheme
of
finite
type
over
O
K
,
whose
generic
fiber
may
be
identified
with
the
[uniquely
39
determined,
up
to
unique
isomorphism]
smooth
compactification
of
X
over
K.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Write
A
⊆
K(X)
for
the
discrete
valuation
ring
associated
to
v.
Note
that
A
may
be
written
as
the
direct
limit
[i.e.,
in
fact,
union]
of
a
direct
system
of
finitely
generated
subalgebras
{A
i
⊆
A}
i∈I
over
O
K
.
Moreover,
since
the
field
extension
K
⊆
K(X)
is
finitely
generated,
and
each
A
i
is
[finitely
generated
over
the
complete
discrete
valuation
ring
O
K
,
hence]
excellent,
we
may
assume
without
loss
of
generality,
i.e.,
by
replacing
A
i
by
its
normalization
in
K(X),
that
each
A
i
is
normal
with
field
of
fractions
equal
to
K(X).
Write
p
i
⊆
A
i
for
the
prime
ideal
determined
by
the
maximal
ideal
of
A.
Thus,
since
v
is
a
p-valuation
[over
K],
it
follows
immediately
that
each
of
the
natural
inclusions
O
K
→
(A
i
)
p
i
→
A
is
a
homomorphism
of
local
rings.
Next,
let
us
observe
that
since
the
residue
field
extension
determined
by
the
natural
inclusion
O
K
⊆
A
of
local
rings
is
assumed
to
be
transcendental,
it
follows
that
there
exists
an
element
i
∈
I
such
that
the
residue
field
k(p
i
)
of
p
i
is
a
transcendental
extension
of
the
residue
field
of
O
K
.
Let
Z
be
a
compactified
model
of
X
over
O
K
that
contains
Spec
A
i
as
an
open
subscheme
[cf.
Proposition
2.3,
(i)].
Then
since
Z
is
of
dimension
2
[cf.
Remark
2.1.1],
it
follows
that
the
height
of
p
i
is
equal
to
1
or
2.
On
the
other
hand,
if
p
i
is
of
height
2,
then
it
follows
that
p
i
corresponds
to
a
closed
point
of
Z
s
,
hence
that
k(p
i
)
is
a
finite
extension
of
the
residue
field
of
O
K
,
i.e.,
in
contradiction
to
our
assumption
of
transcendality.
Thus,
we
conclude
that
p
i
is
of
height
1,
hence
that
(A
i
)
p
i
is
a
discrete
valuation
ring
whose
field
of
fractions
is
equal
to
K(X).
But
this
implies
that
(A
i
)
p
i
=
A.
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
First,
we
observe
that,
after
possibly
replac-
ing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
G-equivariant
finite
morphism
X
→
P
1
O
K
to
the
projective
line
over
O
K
[equipped
with
the
trivial
action
by
G].
Indeed,
since
O
K
is
a
complete
discrete
valuation
ring,
by
deforming
any
[suitably
large
positive
power
of
a]
very
ample
line
bundle
on
the
projective
curve
X
s
,
we
obtain
a
very
ample
line
bundle
L
on
X
,
hence,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
a
pair
of
global
sections
σ
1
,
σ
2
of
the
line
bundle
L
such
that
the
G-orbit
of
the
zero
locus
of
σ
1
is
disjoint
from
the
G-orbit
of
the
zero
locus
of
σ
2
.
Thus,
for
i
=
1,
2,
the
product
σ
i
G
of
the
G-translates
of
σ
i
determines
a
global
section
of
the
[still
very
ample!]
tensor
product
L
G
of
G-translates
of
L
such
that
σ
1
G
and
σ
2
G
still
have
disjoint
zero
loci,
hence
determine
a
G-equivariant
finite
morphism
X
→
P
1
O
K
over
O
K
,
as
desired.
Fix
such
a
finite
morphism,
and
write
f
∈
K(X)
for
the
rational
function
on
X
determined
by
the
standard
coordinate
function
on
P
1
O
K
.
Next,
we
recall
that
it
follows
from
the
stable
reduction
theorem
[cf.
[DM],
[Knud]]
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
we
may
assume
without
loss
of
generality
that
every
closed
point
in
the
support
Supp(f
)
[in
the
smooth
compactification
of
X]
of
the
principal
divisor
associ-
ated
to
f
is
K-rational,
and
that
X
has
stable
reduction
over
K.
Moreover,
by
replacing
X
by
a
suitable
G-stable
open
subscheme
of
X,
we
may
assume
with-
out
loss
of
generality
that
Supp(f
)
is
contained
in
the
set
of
cusps
of
X.
Write
X
†
for
the
compactified
stable
model
of
X
over
O
K
;
E
⊆
X
†
for
the
reduced
40
closed
subscheme
determined
by
the
set
of
closed
points
where
an
irreducible
component
of
the
zero
divisor
of
f
on
X
†
intersects
an
irreducible
component
of
the
divisor
of
poles
of
f
on
X
†
.
Thus,
the
action
of
G
on
X
extends
to
X
†
.
X
†
,x
for
the
completion
of
the
local
ring
of
X
†
at
x.
Fix
a
Let
x
∈
E.
Write
O
uniformizer
π
K
∈
O
K
.
Next,
suppose
that
x
is
a
smooth
closed
point
of
X
†
.
Then
there
exist
nonzero
integers
a,
b
of
opposite
sign
and
a
unit
u
∈
(O
K
[[t]])
×
[where
t
denotes
an
indeterminate],
together
with
an
isomorphism
of
topological
O
K
-algebras
∼
∼
X
†
,x
(
→
X
†
,x
→
O
K
[[t]],
such
that
the
image
of
f
in
the
field
of
fractions
of
O
O
b
O
K
[[t]])
is
of
the
form
u·t
a
·π
K
.
Next,
observe
that,
by
replacing
K
by
a
suitable
finite
extension
field
of
K
[so
it
may
no
longer
be
the
case
that
the
element
“π
K
”
is
a
uniformizer
of
O
K
!],
we
may
assume
without
loss
of
generality
that
there
−b
exists
an
element
γ
∈
O
K
such
that
γ
a
=
π
K
.
Write
x
η
for
the
K-valued
point
of
the
smooth
compactification
of
X
determined
by
the
section
of
the
structure
morphism
X
†
→
Spec
O
K
corresponding
to
the
homomorphism
of
topological
O
K
-algebras
∼
X
†
,x
→)
(
O
O
K
[[t]]
−→
O
K
that
maps
t
→
γ
∈
O
K
;
x
η
for
the
K-valued
point
of
the
smooth
compactifica-
tion
of
X
determined
by
the
section
of
the
structure
morphism
X
†
→
Spec
O
K
corresponding
to
the
homomorphism
of
topological
O
K
-algebras
∼
X
†
,x
→)
O
K
[[t]]
−→
O
K
(
O
that
maps
t
→
0
∈
O
K
;
X
†
[x]
for
the
compactified
stable
model
of
X
\
{x
η
,
x
η
}
over
O
K
.
Thus,
it
follows
immediately
from
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that
the
natural
inclusion
X
\
{x
η
,
x
η
}
→
X
determines
a
natural
birational,
dominant
morphism
X
†
[x]
→
X
†
.
Finally,
we
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
the
rational
function
f
is
a
unit
[at
x
η
,
hence]
at
the
generic
point
of
the
unique
irreducible
component
of
(X
†
[x])
s
that
maps
to
a
closed
point
of
X
s
†
;
in
particular,
the
zero
divisor
of
f
does
not
intersect
the
divisor
of
poles
of
f
in
some
Zariski
neighborhood
of
this
irreducible
component.
Next,
suppose
that
x
is
a
nonsmooth
closed
point
of
X
†
.
Then
since
the
X
†
,x
)
s
are
Q-Cartier
divisors,
it
follows
two
irreducible
components
of
(Spec
O
r
that
there
exist
positive
integers
a,
b,
r
and
a
unit
u
∈
(O
K
[[s,
t]]/(st
−
π
K
))
×
[where
s,
t
denote
indeterminates],
together
with
an
isomorphism
of
topological
∼
r
X
†
,x
→
O
K
-algebras
O
O
K
[[s,
t]]/(st−π
K
),
such
that
the
image
of
some
positive
∼
r
power
of
f
in
the
field
of
fractions
of
O
X
†
,x
(
→
O
K
[[s,
t]]/(st
−
π
K
))
is
of
the
a
−b
form
u·s
·t
.
Next,
observe
that,
by
replacing
K
by
a
suitable
finite
extension
field
of
K
[so
it
may
no
longer
be
the
case
that
the
element
“π
K
”
is
a
uniformizer
of
O
K
!],
we
may
assume
without
loss
of
generality
that
there
exists
an
element
γ
∈
O
K
such
that
γ
a+b
=
π
K
.
Write
x
η
for
the
K-valued
point
of
the
smooth
compactification
of
X
corresponding
to
the
section
of
the
structure
morphism
X
†
→
Spec
O
K
induced
by
the
homomorphism
of
topological
O
K
-algebras
∼
X
†
,x
→)
O
K
[[s,
t]]/(st
−
π
r
)
−→
O
K
(
O
K
41
def
that
maps
s
→
γ
br
∈
O
K
,
t
→
γ
ar
∈
O
K
;
x
η
=
x
η
;
X
†
[x]
for
the
compactified
stable
model
of
X
\
{x
η
,
x
η
}
=
X
\
{x
η
}
over
O
K
.
Thus,
it
follows
immediately
from
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that
the
natural
inclusion
X
\
{x
η
,
x
η
}
=
X
\
{x
η
}
→
X
determines
a
natural
birational,
dom-
inant
morphism
X
†
[x]
→
X
†
.
Finally,
we
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
the
rational
function
f
is
a
unit
[at
x
η
,
hence]
at
the
generic
point
of
the
unique
irreducible
component
of
(X
†
[x])
s
that
maps
to
a
closed
point
of
X
s
†
;
in
particular,
the
zero
divisor
of
f
does
not
in-
tersect
the
divisor
of
poles
of
f
in
some
Zariski
neighborhood
of
this
irreducible
component.
Next,
observe
that
the
underlying
set
of
E
is
finite.
Thus,
by
replacing
X
by
a
suitable
G-stable
open
subscheme
of
X,
we
may
assume
without
loss
of
generality
that
for
each
x
∈
E,
the
K-valued
points
x
η
,
x
η
constructed
above
are
contained
in
the
set
of
cusps
of
X.
Then
it
follows
immediately
from
the
above
discussion
that
the
zero
divisor
of
f
on
X
†
does
not
intersect
the
divisor
of
poles
of
f
on
X
†
.
But
this
implies
that
f
determines
a
G-equivariant
domi-
nant
morphism
X
†
→
P
1
O
K
over
O
K
whose
restriction
to
the
respective
generic
fibers
coincides
with
the
restriction
to
the
respective
generic
fibers
of
the
finite
morphism
X
→
P
1
O
K
over
O
K
constructed
above.
Thus,
since
X
†
is
normal,
we
conclude
that
the
morphism
X
†
→
P
1
O
K
admits
a
factorization
X
†
→
X
→
P
1
O
K
,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(iv).
First,
we
observe
that
since
the
operation
of
forming
the
quotient
of
Y
sst
by
G
commutes
with
flat
base-change,
one
verifies
immediately
that
it
suffices
to
verify
assertion
(iv)
after
performing
any
finite,
faithfully
flat
base-change
from
O
K
to
the
ring
of
integers
in
a
finite
extension
field
of
K.
In
particular,
by
replacing
K
by
a
suitable
finite
extension
field
of
K,
we
may
assume
without
loss
of
generality
that
the
cusps
of
X
are
K-rational,
and
that
X
has
stable
reduction
over
K
[cf.
[DM],
[Knud]].
[In
fact,
these
conditions
are
satisfied
even
if
one
does
not
pass
to
a
finite
extension
field
of
the
original
given
K,
but
we
omit
a
proof
of
this
fact
since
it
is
not
logically
necessary
for
the
present
discussion.]
Write
X
st
for
the
compactified
stable
model
of
X
over
O
K
.
Next,
let
us
observe
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
one
may
regard
Y
sst
as
the
compactified
stable
model
associated
to
the
hyperbolic
curve
Y
obtained
by
removing
from
Y
a
collection
of
G-orbits
of
K-rational
points
of
Y
such
that
the
cardinality
of
the
set
of
G-orbits
of
closed
points
of
Y
s
sst
contained
in
the
intersection
of
any
irreducible
component
of
Y
s
sst
with
the
image
of
the
corresponding
collection
of
O
K
-rational
points
of
Y
sst
is
≥
3.
In
particular,
one
verifies
immediately
that,
by
replacing
Y
by
Y
,
we
may
assume
without
loss
of
generality
that
the
cardinality
of
the
set
of
closed
points
of
each
irreducible
component
of
X
s
that
lie
in
the
image
of
the
cusps
of
Y
sst
is
≥
3.
Next,
let
us
observe
that
it
follows
immediately
from
the
definition
of
the
natural
quotient
morphism
Y
sst
→
X
that
the
natural
morphism
Y
sst
→
X
st
over
O
K
induced
by
the
morphism
Y
→
X
[cf.
[ExtFam],
Theorem
A]
admits
42
a
factorization
Y
sst
−→
X
−→
X
st
,
where
we
note
that
it
follows
immediately
from
the
definition
of
X
that
X
is
a
compactified
model
of
X
over
O
K
.
Thus,
it
follows
immediately
from
the
above
discussion
of
Y
sst
,
together
with
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that
the
morphism
X
→
X
st
is
birational
and
quasi-finite,
hence,
by
Zariski’s
Main
Theorem,
an
isomorphism.
In
particular,
we
conclude
that
X
is
a
compactified
semistable
model
of
X
over
O
K
.
Moreover,
it
follows
immediately
from
Remark
2.1.3
that
the
natural
morphism
Y
s
sst
→
X
s
maps
smooth
points
of
Y
s
sst
to
smooth
points
of
X
s
.
Next,
let
e
Y
∈
Y
s
sst
be
a
node.
Write
e
X
∈
X
s
for
the
image
of
e
Y
via
the
natural
morphism
Y
s
sst
→
X
s
;
G
e
Y
⊆
G
for
the
stabilizer
of
e
Y
in
G;
Y
e
sst
for
the
Y
spectrum
of
the
completion
of
the
local
ring
of
Y
sst
at
e
Y
;
X
e
X
for
the
spectrum
of
the
completion
of
the
local
ring
of
X
at
e
X
.
Thus,
G
e
Y
acts
naturally
on
Y
e
sst
;
X
e
X
may
be
identified
with
the
quotient
of
Y
e
sst
by
the
action
of
G
e
Y
;
the
Y
Y
set
B
Y
of
irrreducible
components
of
(Y
e
sst
)
may
be
identified
with
the
set
[of
Y
s
cardinality
2]
of
branches
of
e
Y
;
the
set
B
X
of
irrreducible
components
of
(X
e
X
)
s
is
of
cardinality
1
if
and
only
if
e
X
is
a
smooth
point
of
X
s
and
may
be
identified
with
the
set
[of
cardinality
2]
of
branches
of
e
X
whenever
e
X
is
a
node.
On
the
other
hand,
it
follows
immediately
from
elementary
commutative
algebra
that
the
set
B
X
may
be
naturally
identified
with
the
set
of
G
e
Y
-orbits
of
B
Y
.
The
remaining
portion
of
assertion
(iv)
now
follows
formally.
This
completes
the
proof
of
assertion
(iv).
Write
Next,
we
verify
assertions
(v)
and
(vi).
Let
c
∈
VE(
X).
def
R
c
=
lim
O
Z,z
c
,
−→
Z
where
•
Z
ranges
over
the
compactified
semistable
models
with
split
reduction
of
the
domain
curves
of
connected
finite
étale
coverings
Z
→
X
equipped
→
Z
→
X;
with
a
factorization
X
•
c
Z
denotes
the
irreducible
component
or
node
of
Z
s
determined
by
c;
•
z
c
denotes
the
generic
point
of
the
intersection
of
the
[closed
irreducible]
images
in
Z
s
of
the
c
Z
†
associated
to
compactified
semistable
models
with
split
reduction
of
domain
curves
of
connected
finite
étale
coverings
→
Z
†
→
Z
→
X
such
that
the
Z
†
→
X
equipped
with
a
factorization
X
†
morphism
Z
→
Z
extends
to
a
morphism
Z
†
→
Z;
•
the
transition
maps
in
the
direct
limit
are
the
homomorphisms
of
local
rings
induced
by
the
corresponding
scheme-theoretic
morphisms
of
com-
pactified
semistable
models
[which
form
a
directed
inverse
system
—
cf.
Proposition
2.3,
(iii)].
43
Then
it
follows
immediately
from
the
various
definitions
involved
that
the
field
that
O
K
⊆
R
c
,
and
that
R
c
is
a
local
of
fractions
of
R
c
coincides
with
K(
X),
c
⊆
K(
X)
be
a
valuation
domain
whose
maximal
ideal
m
R
c
contains
p.
Let
R
c
for
the
ring
that
dominates
R
c
[cf.,
e.g.,
[EP],
Theorem
3.1.1].
Write
m
R
c
⊆
R
c
.
Thus,
since
O
K
⊆
O
⊆
R
c
⊆
R
c
and
p
∈
m
R
⊆
m
,
we
maximal
ideal
of
R
c
K
R
c
c
,
i.e.,
that
the
valuation
determined
by
the
valuation
conclude
that
O
K
=
K
∩
R
c
is
a
p-valuation.
ring
R
c
may
be
written
as
the
direct
limit
[i.e.,
in
fact,
union]
of
a
direct
Note
that
R
c
}
i∈I
over
O
K
.
Moreover,
system
of
finitely
generated
subalgebras
{R
i
⊆
R
since
R
i
is
[finitely
generated
over
the
complete
discrete
valuation
ring
O
K
,
hence]
excellent,
by
replacing
R
i
by
its
normalization
in
the
subfield
of
K(
X)
generated
by
the
field
of
fractions
of
R
i
and
some
suitable
finite
extension
field
of
K,
we
may
assume
without
loss
of
generality
that
R
i
is
normal,
and
that
there
exists
a
compactified
semistable
model
with
split
reduction
Z
i
of
the
domain
curve
of
a
connected
finite
étale
covering
Z
i
→
X
equipped
with
a
→
Z
i
→
X
such
that
Spec
R
i
arises
as
an
open
subscheme
of
factorization
X
def
Z
i
[cf.
Proposition
2.3,
(i),
(iii)].
Write
p
i
=
m
R
c
∩
R
i
⊆
R
i
.
Thus,
if
we
write
z
i
for
the
point
of
Z
i
that
corresponds
to
p
i
,
then
it
holds
that
c
=
lim
O
Z
,z
.
R
i
i
−→
i∈I
On
the
other
hand,
observe
that,
since
m
R
c
∩
R
c
=
m
R
c
,
it
follows
immediately
from
the
various
definitions
involved
that
each
“O
Z
i
,z
i
(=
(R
i
)
p
i
)”
of
the
above
direct
limit
appears
as
one
of
the
“O
Z,z
c
”
in
the
direct
limit
used
to
define
R
c
.
c
,
hence
that
R
c
=
R
c
,
i.e.,
that
R
c
c
⊆
R
c
⊆
R
In
particular,
we
conclude
that
R
is
the
valuation
ring
associated
to
a
p-valuation.
Thus,
in
summary,
we
obtain
a
natural
map
−→
p-valuations
on
K(
X)
VE(
X)
c
→
R
c
.
Moreover,
one
verifies
immediately
that
this
map
that
maps
VE(
X)
defines
an
inverse
to
the
natural
map
p-valuations
on
K(
X)
−→
VE(
X)
in
the
statement
of
assertion
(v),
hence
that
both
of
these
maps
are
bijective.
Since
both
of
these
maps
are
manifestly
compatible
with
specialization/generization,
we
thus
conclude
that
these
induce
a
bijection
primitive
p-valuations
on
K(
X)
∼
prim
→
VE(
X)
as
in
the
statement
of
assertion
(v).
Thus,
to
complete
the
proof
of
assertion
(v),
it
suffices
to
verify
that
the
natural
map
X(Ω)
−→
p-valuations
on
K(
X)
44
[i.e.,
that
assigns
to
an
element
of
X(Ω)
the
associated
point-theoretic
valuation
on
K(
X)]
is
injective.
To
this
end,
let
x̃
∈
X(Ω).
Write
v
for
the
point-theoretic
valuation
on
K(
X)
associated
to
x̃.
Then
observe
that
x̃
is
defined
over
K
if
and
is
strict.
If
this
inclusion
is
strict,
then
the
only
if
the
inclusion
O
v
·
K
⊆
K(
X)
is
the
valuation
ring
determined
[i.e.,
in
the
usual
sense
subring
O
v
·
K
⊆
K(
X)
of
the
classical
theory
of
one-dimensional
function
fields
over
algebraically
closed
is
strict,
the
fields]
by
x̃.
In
particular,
whenever
the
inclusion
O
v
·
K
⊆
K(
X)
point
x̃
∈
X(K)
(⊆
X(Ω))
is
completely
determined
by
v.
Thus,
it
remains
to
In
this
case,
the
valuation
v
is
real,
and
consider
the
case
where
O
v
·K
=
K(
X).
induces,
by
passing
to
the
respective
completions,
an
the
inclusion
K
⊆
K(
X)
with
respect
to
v.
In
particular,
isomorphism
of
Ω
with
the
completion
of
K(
X)
we
obtain
a
natural
homomorphism
K(
X)
→
Ω,
which
completely
determines
the
point
x̃
∈
X(Ω).
This
completes
the
proof
of
assertion
(v).
Assertion
(vi)
follows
immediately
from
the
proof
of
assertion
(v).
Next,
we
verify
assertion
(vii).
Let
x
∈
X(Ω).
Write
v
for
the
point-theoretic
valuation
on
K(
X)
associated
to
x;
φ
x
:
O
v
→
O
Ω
for
the
homomorphism
obtained
by
evaluating
rational
functions
at
x.
Let
q
⊆
O
v
be
a
prime
ideal
that
contains
p.
Then
observe
that
it
follows
immediately
from
the
construction
of
v
that
p
1
·Ker(φ
x
)
⊆
Ker(φ
x
).
Since
p
∈
q,
we
thus
conclude
that
Ker(φ
x
)
⊆
q.
On
the
other
hand,
observe
that
[it
follows
immediately
from
the
construction
of
v
that]
this
inclusion
implies
that
q
contains
[hence
coincides
with]
the
radical
of
the
ideal
(p,
Ker(φ
x
))
⊆
O
v
,
which
is
easily
seen
to
be
equal
to
the
maximal
ideal
m
v
of
O
v
.
Thus,
we
conclude
that
v
is
primitive.
Let
a
∈
m
v
.
Then
since
v
is
real,
Next,
let
v
be
a
real
p-valuation
on
K(
X).
there
exists
a
positive
integer
N
such
that
a
N
∈
(p).
In
particular,
any
prime
ideal
that
contains
p
contains
[hence
coincides
with]
m
v
.
Thus,
we
conclude
that
v
is
primitive.
For
each
z
∈
O
v
,
write
Next,
let
v
be
a
primitive
p-valuation
on
K(
X).
for
the
O
v
-subalgebra
generated
by
1
.
Thus,
if
(K
⊆)
(O
v
)
p
=
(O
v
)
z
⊆
K(
X)
z
then
it
follows
immediately
from
the
classical
theory
of
one-dimensional
K(
X),
function
fields
over
algebraically
closed
fields
that
v
is
a
point-theoretic
valuation.
Note
Therefore,
we
may
assume
without
loss
of
generality
that
(O
v
)
p
=
K(
X).
that
this
implies
that
for
each
x
∈
O
v
\
{0},
there
exist
a
positive
integer
N
and
y
∈
O
v
such
that
p
N
=
xy.
Moreover,
in
this
situation,
it
holds
that
m
K
O
v
=
m
v
.
Indeed,
since
v
is
a
p-valuation,
the
inclusion
m
K
O
v
⊆
m
v
is
immediate.
Now
suppose
that
there
exists
an
element
x
∈
m
v
\
m
K
O
v
.
Then
it
follows
that
1
p
∈
(O
v
)
x
,
hence
that
there
exists
a
prime
ideal
p
v
of
O
v
such
that
x
∈
p
v
,
and
p
∈
p
v
.
On
the
other
hand,
since
v
is
primitive,
we
conclude
that
x
∈
m
v
=
p
v
,
a
contradiction.
This
completes
the
proof
of
the
equality
in
the
above
display.
Note
that
this
equality
implies
that
for
each
x
∈
m
v
\
{0},
there
exist
a
positive
integer
N
and
y
∈
O
v
such
that
x
N
=
py.
In
particular,
it
follows
immediately
from
the
various
definitions
involved
that
v
coincides
with
the
real
valuation
45
determined
by
the
assignment
O
v
\
{0}
x
→
sup{
∈
Q
|
x
∈
p
·
O
v
}
=
inf{
∈
Q
|
x
−1
∈
p
−
·
O
v
}
∈
R.
This
completes
the
proof
of
assertion
(vii).
Next,
we
verify
assertion
(viii).
First,
let
us
observe
that
it
follows
immedi-
ately
from
the
final
portion
of
Proposition
2.3,
(vii),
that
the
natural
map
an
−→
VE(
X)
X
the
center-chain
associated
to
the
—
i.e.,
that
assigns
to
a
valuation
on
K(
X)
valuation
—
admits
a
factorization
∼
an
→
prim
−→
VE(
X),
X
VE(
X)
ι
X
where
the
first
arrow
is
a
bijection,
and
the
second
arrow
ι
X
denotes
the
natural
inclusion.
On
the
other
hand,
it
follows
immediately
from
the
discussion
of
the
tor
”
in
Definition
ratios
“ρ
b,v
”
in
Remark
2.1.4
and
the
construction
of
“VE(
X)
also
admits
a
factorization
an
→
VE(
X)
2.2,
(vi),
that
this
natural
map
X
tor
−→
VE(
X),
an
−→
VE(
X)
X
θ
X
X
where
the
first
map
θ
X
is
defined
by
considering
ratios
“ρ
b,v
”
as
in
Remark
2.1.4
tor
”
in
Definition
2.2,
(vi)],
and
the
second
[cf.
also
the
construction
of
“VE(
X)
arrow
X
is
the
natural
map
discussed
in
the
final
portion
of
Definition
2.2,
(vi).
In
particular,
we
obtain
a
commutative
diagram
of
maps
of
sets
an
X
⏐
⏐
tor
−−−−→
VE(
X)
θ
X
⏐
⏐
X
prim
−−−−→
VE(
X).
VE(
X)
ι
X
Note
that
the
commutativity
of
the
diagram
already
implies
that
θ
X
is
injective.
Moreover,
one
verifies
immediately
—
i.e.,
by
considering
suitable
“v”
as
in
Remark
2.1.4
—
that
each
composite
map
θ
Z
tor
−→
VE(Z)
tor
an
−→
VE(
X)
X
θ
X
—
where
the
second
arrow
is
the
natural
projection
arising
from
the
inverse
limit
tor
[cf.
Definition
2.2,
(vi)]
—
has
dense
image.
Thus,
in
the
definition
of
VE(
X)
an
[cf.
[Brk],
Theorem
1.2.1],
together
it
follows
from
the
compactness
of
X
with
the
easily
verified
fact
[cf.
the
construction
of
Definition
2.2,
(vi)]
that
VE(Z)
tor
is
Hausdorff,
that
to
verify
that
θ
X
is
a
homeomorphism,
it
suffices
to
verify
that
each
map
θ
Z
is
continuous.
Moreover,
once
one
knows
that
θ
X
is
a
homeomorphism,
one
may
construct
a
natural
splitting
τ
X
as
in
the
statement
46
of
Proposition
2.3,
(viii),
by
constructing
a
natural
splitting
of
the
natural
→
VE(
X)
prim
inclusion
ι
X
.
On
the
other
hand,
such
a
natural
splitting
VE(
X)
of
ι
X
is
implicit
in
the
content
of
Proposition
2.2,
(x)
[which
will
be
verified
below,
independently
of
the
present
assertion
(viii)],
i.e.,
one
assigns
to
each
the
unique
generization
∈
VE(
X)
prim
of
c.
nonprimitive
element
c
∈
VE(
X)
Thus,
in
summary,
to
complete
the
proof
of
assertion
(viii),
it
suffices
to
verify
that
each
map
an
−→
VE(Z)
tor
θ
Z
:
X
as
in
the
above
discussion
is
continuous.
Let
Z
†
be
a
toral
compactified
semistable
model
relative
to
Z.
Then
we
shall
refer
to
an
open
subscheme
U
of
Z
s
†
as
a
componentwise
open
of
Z
†
if
U
is
an
open
subscheme
of
Z
s
†
whose
underlying
open
subset
is
the
complement
of
a
node
or
an
irreducible
component
of
Z
s
†
.
Observe
that
it
follows
immediately
from
the
construction
of
VE(Z)
tor
given
in
Definition
2.2,
(vi),
that
each
componentwise
open
of
each
toral
compactified
semistable
model
relative
to
Z
determines,
in
a
natural
way,
a
closed
subset
of
VE(Z)
tor
.
We
shall
refer
to
the
closed
subsets
of
VE(Z)
tor
obtained
in
this
way
as
componentwise
closed
subsets
of
VE(Z)
tor
.
Note
that
it
follows
immediately
from
the
construction
of
VE(Z)
tor
given
in
Definition
2.2,
(vi),
that
the
com-
plements
of
the
componentwise
closed
subsets
of
VE(Z)
tor
form
an
open
basis
of
the
topology
of
VE(Z)
tor
.
Thus,
to
complete
the
proof
of
the
contininuity
of
θ
Z
,
it
suffices
to
verify
that
the
inverse
image
via
θ
Z
of
any
componentwise
an
.
But
this
follows
immediately
from
closed
subset
of
VE(Z)
tor
is
closed
in
X
the
definition
of
the
topology
of
the
Berkovich
spaces
[cf.
the
discussion
of
[Brk],
an
”
in
the
§1.1,
§1.2]
that
appear
in
the
inverse
limit
that
is
used
to
define
“
X
statement
of
Proposition
2.3,
(vii).
This
completes
the
proof
of
assertion
(viii).
Next,
we
verify
assertion
(ix).
In
the
following,
we
assume
that
δ(c
1
,
c
2
)
=
+∞.
Let
us
first
consider
the
case
where
δ(c
1
,
c
2
)
≥
1.
Then
it
follows
immediately
from
the
definition
of
δ(−,
−)
that
there
exists
a
compactified
semistable
model
Z
with
split
reduction
of
a
connected
finite
étale
covering
Z
→
X
equipped
→
Z
→
X
such
that
δ(z
c
,
z
c
)
≥
1
[cf.
the
notation
with
a
factorization
X
1
2
of
Proposition
2.3,
(vi)].
In
particular,
there
exists
a
node
e
of
Z
s
that
does
not
coincide
with
z
c
1
or
z
c
2
,
and
whose
corresponding
edge
lies
on
a
path
of
minimal
length
between
z
c
1
and
z
c
2
.
On
the
other
hand,
by
considering
suitable
torally
compactified
semistable
models
relative
to
Z
at
e,
we
conclude
that
δ(c
1
,
c
2
)
=
+∞,
in
contradiction
to
our
assumption
that
δ(c
1
,
c
2
)
=
+∞.
Thus,
to
complete
the
proof
of
assertion
(ix),
it
suffices
to
consider
the
case
where
δ(c
1
,
c
2
)
=
0.
Let
us
first
observe
that
the
condition
that
δ(c
1
,
c
2
)
=
0
such
that
c
3
c
1
and
c
3
c
2
.
implies
the
existence
of
an
element
c
3
∈
VE(
X)
Finally,
we
verify
the
uniqueness
of
such
an
element
c
3
∈
VE(
X).
Let
c
4
∈
VE(
X)
be
such
that
c
3
=
c
4
,
c
4
c
1
,
and
c
4
c
2
.
Then
since
δ(c
3
,
c
4
)
<
+∞,
it
follows
immediately
from
the
above
discussion
that
there
exists
an
element
such
that
c
5
c
3
,
and
c
5
c
4
.
Moreover,
since
c
1
=
c
2
,
and
c
5
∈
VE(
X)
c
3
=
c
4
,
by
permuting
{c
3
,
c
4
}
or
{c
1
,
c
2
}
if
necessary,
we
may
assume
without
loss
of
generality
that
c
5
=
c
3
,
and
c
3
=
c
1
.
On
the
other
hand,
the
resulting
47
nontriviality
of
the
specialization
relations
c
5
c
3
c
1
then
contradicts
the
1-dimensionality
of
the
special
fibers
of
the
compactified
semistable
models
“Z”
This
completes
the
proof
of
assertion
that
appear
in
the
definition
of
VE(
X).
(ix).
def
Next,
we
verify
assertion
(x).
Write
c
1
=
c.
Suppose
that
c
2
c
1
,
c
2
c
1
\
{c
1
}.
Then
it
follows
that
δ(c
2
,
c
)
<
for
distinct
elements
c
2
,
c
2
∈
VE(
X)
2
+∞.
Thus,
we
conclude
from
Proposition
2.3,
(ix),
that
there
exists
an
element
such
that
c
3
c
2
,
and
c
3
c
.
In
particular,
by
permuting
c
3
∈
VE(
X)
2
{c
2
,
c
2
}
if
necessary,
we
may
assume
without
loss
of
generality
that
c
3
=
c
2
,
and
c
2
=
c
1
.
On
the
other
hand,
the
resulting
nontriviality
of
the
specialization
relations
c
3
c
2
c
1
then
contradicts
the
uniqueness
portion
of
Proposition
2.3,
(ix).
This
completes
the
proof
of
assertion
(x).
Next,
we
verify
assertion
(xi).
Let
Z
→
X
be
a
[connected]
finite
étale
Ga-
→
Z
→
X
such
that
Z
has
split
lois
covering
equipped
with
a
factorization
X
∗
stable
reduction
over
K;
Z
a
compactified
semistable
model
with
split
reduc-
tion
of
Z
over
O
K
that
is
stabilized
by
the
natural
action
of
Gal(Z/X).
[Note
that
it
follows
immediately
from
Proposition
2.3,
(iii),
that
such
compactified
semistable
models
form
a
directed
inverse
system
that
is
cofinal
in
the
directed
inverse
system
that
appears
in
the
definition
of
VE(
X).]
Write
Z
for
the
com-
pactified
stable
model
with
split
reduction
of
Z
over
O
K
;
Γ
for
the
dual
graph
of
Z
s
;
Γ
∗
for
the
dual
graph
of
Z
s
∗
.
Observe
that
the
natural
action
of
s(H)
on
Γ
∗
factors
through
a
finite
quotient
of
s(H).
Thus,
it
follows
immediately
from
[CbTpIV],
Corollary
1.15,
(iii),
that
the
natural
action
of
s(H)
on
Γ
has
a
fixed
point
c
∈
Γ.
On
the
other
hand,
it
follows
immediately
from
the
well-known
theory
of
stable
and
semistable
models
[i.e.,
which
may
be
reduced,
by
adding
finitely
many
suitably
positioned
cusps,
to
the
theory
of
pointed
stable
curves
and
contraction
morphisms
that
arise
from
eliminating
cusps,
as
exposed
in
[Knud]]
that
the
inverse
image
of
[the
node
or
interior
of
an
irreducible
compo-
nent
in
Z
s
∗
corresponding
to]
c
via
the
dominant
morphism
Z
∗
→
Z
determines
a
tree
inside
Γ
∗
.
Moreover,
we
recall
that
any
action
of
a
finite
group
on
a
tree
has
a
fixed
point
[cf.,
e.g.,
[SemiAn],
Lemma
1.8,
(ii)].
Thus,
we
conclude
that
the
natural
action
of
s(H)
on
Γ
∗
has
a
fixed
point.
Since
any
inverse
limit
of
nonempty
finite
sets
is
nonempty,
we
thus
conclude
that
the
natural
action
of
has
a
fixed
point
∈
VE(
X),
hence
from
Proposition
2.3,
(x),
s(H)
on
VE(
X)
prim
.
that
the
natural
action
of
s(H)
on
VE(
X)
prim
has
a
fixed
point
∈
VE(
X)
This
completes
the
proof
of
assertion
(xi).
Next,
we
verify
assertion
(xii).
Fix
a
prime
number
l
∈
Σ\{p}.
Then
observe
that
it
follows
from
the
stable
reduction
theorem
[cf.
[DM],
[Knud]]
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
we
may
assume
without
loss
of
generality
that
Σ
=
{p,
l},
and
that
X
has
stable
reduction
over
K.
Write
X
for
the
[unique,
up
to
unique
isomorphism]
compactified
stable
model
of
X
over
O
K
.
Next,
observe
that
it
follows
immediately
from
Hurwitz’s
formula,
together
with
the
well-known
structure
of
geometric
fundamental
groups
of
hyperbolic
curves
over
fields
of
characteristic
zero
[cf.,
e.g.,
[CmbGC],
Remark
1.1.3],
that,
48
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
connected
geometrically
pro-p
finite
étale
covering
Y
→
X
of
hyperbolic
curves
with
split
stable
reduction
over
K
that
satisfies
the
condition
that
Y
is
of
genus
g
Y
≥
2.
Write
Y
†
for
the
smooth
compactification
of
Y
over
K;
Y
†
for
the
[unique,
up
to
unique
isomorphism]
compactified
stable
model
of
Y
†
over
O
K
.
Then
observe
that,
if
Y
†
is
smooth
over
O
K
,
then
it
follows
immediately
from
the
non-injectivity
of
the
natural
surjective
homomorphism
ab
Π
ab
Y
†
⊗
Z/pZ
Π
Y
†
⊗
Z/pZ
s
[where
we
recall
that,
since
Y
†
is
of
genus
g
Y
†
≥
2,
the
domain
of
this
homo-
morphism
is
of
cardinality
p
2g
Y
†
,
while
the
codomain
of
this
homomorphism
is
of
cardinality
≤
p
g
Y
†
],
together
with
Hurwitz’s
formula
[cf.
also
Zariski-Nagata
purity;
[ExtFam],
Theorem
A],
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
[connected]
finite
étale
cyclic
covering
Y
‡
→
Y
†
of
hyperbolic
curves
over
K
that
is
of
degree
p
and,
moreover,
satisfies
the
property
that
Y
‡
has
bad
reduction.
In
particular,
by
replacing
Y
‡
×
Y
†
Y
by
Y
,
we
may
assume
without
loss
of
generality
that
Y
has
bad
reduction.
More-
over,
by
replacing
the
connected
geometrically
pro-p
finite
étale
covering
Y
→
X
by
its
Galois
closure,
we
may
assume
without
loss
of
generality
that
Y
→
X
is
a
[connected]
geometrically
pro-p
finite
étale
Galois
covering
[cf.
Remark
2.1.3;
Hurwitz’s
formula].
Next,
observe
that
it
follows
immediately
from
the
theory
of
admissible
cov-
erings
[cf.,
e.g.,
[Hur],
§3],
together
with
Hurwitz’s
formula
[and
the
well-known
structure
of
geometric
pro-l
fundamental
groups
of
hyperbolic
curves
over
fields
of
characteristic
p
=
l
—
cf.,
e.g.,
[CmbGC],
Remark
1.1.3],
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
[connected]
geometrically
pro-l
finite
étale
Galois
covering
Z
→
Y
of
hyperbolic
curves
with
split
stable
reduction
over
K
that
satisfies
the
condition
that
every
irreducible
component
of
the
special
fiber
of
the
[unique,
up
to
unique
isomorphism]
sta-
ble
model
of
Z
is
a
smooth
curve
of
genus
≥
2.
Here,
note
that,
by
replacing
Z
→
Y
by
the
composite
of
the
Gal(Y
/X)-conjugates
of
the
admissible
covering
Z
→
Y
,
we
may
assume
without
loss
of
generality
that
the
composite
covering
def
Z
→
Y
→
X
is
Galois.
Thus,
by
taking
X
†
=
Z,
we
obtain
a
[connected]
geometrically
pro-Σ
finite
étale
Galois
covering
X
†
→
X
of
hyperbolic
curves
satisfying
the
conditions
in
the
statement
of
assertion
(xii),
as
desired.
This
completes
the
proof
of
assertion
(xii),
hence
of
Proposition
2.3.
Remark
2.3.1.
We
maintain
the
notation
of
Proposition
2.3.
Then
we
observe
that
the
statement
of
Proposition
2.3,
(ii),
becomes
false
if
one
omits
the
con-
dition
that
the
p-valuation
v
is
residue-transcendental.
Indeed,
it
suffices
to
construct
an
example
of
a
discrete
p-valuation
on
K(X)
whose
residue
field
is
algebraic
over
the
residue
field
of
O
K
.
Suppose
that
no
finite
extension
field
of
the
residue
field
of
O
K
is
separably
closed
[a
condition
that
is
satisfied
if,
for
instance,
the
residue
field
of
O
K
is
finite].
Then
one
verifies
immediately
that
49
ur
)
\
X(K)
=
∅,
and
that
for
any
x
∈
X(
K
ur
)
\
X(K),
the
point-theoretic
X(
K
valuation
associated
to
x
on
K(X)
satisfies
the
desired
properties.
Remark
2.3.2.
An
alternative
proof
of
Proposition
2.3,
(iv),
may
be
found
in
[Ray2],
Proposition
5.
The
proof
of
Proposition
2.3,
(iv),
given
in
the
present
paper
is
of
interest
in
that
it
involves
techniques
that
are
closer
to
the
overall
approach
of
the
present
paper.
∼
an
→
tor
of
Proposition
2.3,
(viii),
Remark
2.3.3.
The
homeomorphism
X
VE(
X)
is
essentially
the
same
as
the
homeomorphism
of
[Lpg1],
Proposition
1.1,
but
we
give
[essentially]
self-contained
statements
and
proofs
here
in
the
language
of
the
present
discussion
for
the
sake
of
completeness.
Remark
2.3.4.
We
maintain
the
notation
of
Proposition
2.3.
Let
X
be
a
com-
pactified
semistable
model
of
X
over
O
K
;
x
∈
X
a
smooth
closed
point.
Write
η
for
the
generic
point
of
the
unique
irreducible
component
of
X
s
that
contains
x.
Then
one
may
construct
a
p-valuation
v
on
K(X)
associated
to
x
by
taking
the
ring
of
integers
O
v
to
consist
of
the
elements
∈
K(X)
that
are
integral
with
respect
to
the
discrete
valuation
on
K(X)
associated
to
η
and,
moreover,
map
to
an
element
in
the
residue
field
k(η)
of
X
at
η
that
is
integral
with
respect
to
the
discrete
valuation
on
k(η)
determined
by
x.
Note
that
η
determines
a
prime
ideal
of
O
v
that
contains
p.
In
particular,
v
is
nonprimitive.
Proposition
2.4
(First
properties
of
resolution
of
nonsingularities).
Let
Σ
⊆
Primes
be
a
nonempty
subset;
K
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
hyperbolic
curve
over
K.
Then:
(i)
Let
U
⊆
X
be
an
open
subscheme
[so
U
is
a
hyperbolic
curve
over
K].
Suppose
that
X
satisfies
Σ-RNS.
Then
it
holds
that
U
satisfies
Σ-RNS.
(ii)
Let
f
:
Y
→
X
be
a
connected
geometrically
pro-Σ
finite
étale
covering
over
K
[so
Y
is
a
hyperbolic
curve
over
a
finite
extension
field
of
K].
Then
it
holds
that
X
satisfies
Σ-RNS
if
and
only
if
Y
satisfies
Σ-RNS.
(iii)
Suppose
that
X
satisfies
the
following
condition:
Let
X
be
a
compactified
model
of
X
over
O
K
;
x
∈
X
s
a
closed
point.
Then,
after
possibly
replacing
K
by
a
suitable
finite
ex-
tension
field
of
K,
there
exist
•
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
Y
→
X
of
hyperbolic
curves
over
K,
•
a
compactified
semistable
model
Y
of
Y
over
O
K
,
•
a
morphism
Y
→
X
of
compactified
models
over
O
K
that
restricts
to
the
finite
étale
Galois
covering
Y
→
X,
50
•
an
irreducible
component
D
of
Y
s
whose
normalization
is
of
genus
≥
1,
and
whose
image
in
X
s
is
x
∈
X
s
.
Then
X
satisfies
Σ-RNS.
(iv)
Suppose
that
X
satisfies
Σ-RNS.
Let
X
be
a
compactified
model
of
X
over
O
K
.
Then,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
Y
→
X
over
K,
together
with
a
compactified
stable
model
Y
of
Y
over
O
K
,
such
that
the
covering
Y
→
X
extends
to
a
morphism
Y
→
X
.
(v)
Suppose
that
we
are
in
the
situation
of
Proposition
2.3,
(v),
and
that
X
satisfies
Σ-RNS.
Write
st
,
VE(
X)
st,tor
,
VE(
X)
st,prim
,
VE(
X)
st,pt-th
VE(
X)
VE(
X)
tor
,
VE(
X)
prim
,
VE(
X)
pt-th
for
the
modified
versions
of
VE(
X),
obtained
by
requiring
that
the
compactified
semistable
models
“Z”
that
appear
in
the
inverse
limits
used
to
define
these
sets
be
compactified
stable
models.
[Here,
we
observe
that,
in
light
of
(iv),
the
various
toral
compactified
semistable
models
“Z
†
”
relative
to
“Z”
that
appear
in
the
construction
of
“VE(Z)
tor
”
in
Definition
2.2,
(vi),
may
be
understood
as
being
obtained
as
the
result
of
contracting
suitable
irreducible
components
in
the
special
fibers
[cf.
Remark
2.1.6]
of
suitable
quotients
of
compactified
stable
models
as
in
Proposition
2.3,
(iv).]
Then
the
natural
maps
−→
VE(
X)
st
VE(
X)
tor
−→
VE(
X)
st,tor
VE(
X)
prim
−→
VE(
X)
st,prim
VE(
X)
pt-th
−→
VE(
X)
st,pt-th
VE(
X)
are
bijective.
(vi)
Suppose
that
we
are
in
the
situation
of
Proposition
2.3,
(v).
Then
X
it
holds
that
satisfies
Σ-RNS
if
and
only
if
for
each
c
∈
VE(
X),
R
c
=
lim
O
Z
st
,z
c
,
−→
st
Z
denotes
the
valuation
ring
of
the
p-valuation
associ-
where
R
c
⊆
K(
X)
ated
to
c
[cf.
Proposition
2.3,
(v)];
the
direct
limit
ranges
over
the
set
of
compactified
stable
models
with
split
reduction
Z
st
of
the
domain
curves
of
connected
finite
étale
coverings
Z
→
X
equipped
with
a
factorization
→
Z
→
X;
z
c
denotes
the
center
on
Z
st
determined
by
R
c
;
the
tran-
X
sition
maps
in
the
direct
limit
are
induced
by
the
corresponding
scheme-
theoretic
morphisms
of
compactified
stable
models
[which,
in
light
of
(iv),
form
a
directed
inverse
system].
51
(vii)
Suppose
that
we
are
in
the
situation
of
Proposition
2.3,
(v),
and
that
X
satisfies
Σ-RNS.
Let
l
∈
Σ
\
{p};
H
⊆
G
K
a
closed
subgroup
such
that
the
intersection
H
∩
I
K
of
H
with
the
inertia
subgroup
I
K
of
G
K
(Σ)
admits
a
surjection
to
[the
profinite
group]
Z
l
;
s
:
H
→
Π
X
a
section
(Σ)
of
the
restriction
to
H
of
the
natural
surjection
Π
X
G
K
.
Then
there
prim
that
is
fixed
by
the
restriction,
exists
at
most
one
element
c
∈
VE(
X)
(Σ)
prim
⊆
VE(
X);
if,
via
s,
to
H
of
the
natural
action
of
Π
X
on
VE(
X)
moreover,
the
restriction
to
H
of
the
l-adic
cyclotomic
character
of
K
prim
.
In
has
open
image,
then
there
exists
a
unique
such
element
c
∈
VE(
X)
an
particular,
if
X
is
proper,
then
there
exists
at
most
one
element
c
an
∈
X
[cf.
Proposition
2.3,
(vii)]
that
is
fixed
by
the
restriction,
via
s,
to
H
of
(Σ)
an
;
if,
the
natural
action
of
Π
X
on
the
topological
pro-Berkovich
space
X
moreover,
the
restriction
to
H
of
the
l-adic
cyclotomic
character
of
an
.
K
has
open
image,
then
there
exists
a
unique
such
element
c
an
∈
X
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
various
definitions
in-
volved.
Next,
we
verify
assertion
(iii).
Let
v
be
a
discrete
residue-transcendental
p-valuation
on
K(X).
Then
it
follows
immediately
from
Proposition
2.3,
(ii),
(iii),
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
compactified
semistable
model
X
of
X
over
O
K
such
that
v
arises
from
an
irreducible
component
of
X
s
.
Let
{x
1
,
.
.
.
,
x
N
}
⊆
X
s
be
a
fi-
nite
set
of
distinct
closed
points
in
the
smooth
locus
of
X
s
such
that
every
irreducible
component
of
X
s
whose
normalization
is
of
genus
0
contains
three
points
∈
{x
1
,
.
.
.
,
x
N
}.
Then
it
follows
immediately
from
our
assumption
on
X
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
for
each
positive
integer
i
≤
N
,
there
exist
•
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
Y
i
→
X
over
K,
•
a
morphism
f
i
:
Y
i
→
X
of
compactified
semistable
models
over
O
K
that
restricts
to
the
finite
étale
Galois
covering
Y
i
→
X,
•
an
irreducible
component
D
i
of
(Y
i
)
s
whose
normalization
is
of
genus
≥
1,
and
whose
image
in
X
s
is
x
i
.
Write
f
η
:
Y
→
X
for
the
connected
geometrically
pro-Σ
finite
étale
Galois
covering
over
K
obtained
by
forming
the
composite
of
the
finite
étale
Galois
coverings
{Y
i
→
X}
1≤i≤N
over
K.
Then
it
follows
immediately
from
Propo-
sition
2.3,
(iii),
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
compactified
semistable
model
Y
†
of
Y
over
O
K
that
dominates
the
respective
normalizations
of
the
semistable
models
{Y
i
}
1≤i≤N
in
the
function
field
of
Y
.
In
particular,
for
each
positive
integer
i
≤
N
,
there
exists
an
irreducible
component
D
i
†
of
(Y
†
)
s
whose
normalization
is
of
genus
≥
1,
and
whose
image
in
X
s
is
x
i
.
Next,
let
us
observe
that
it
follows
immediately
from
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that,
after
possibly
replacing
K
by
a
suit-
able
finite
extension
field
of
K,
we
may
regard
Y
†
as
the
compactified
stable
52
model
associated
to
the
hyperbolic
curve
Y
†
obtained
by
removing
from
Y
a
collection
of
K-rational
points
of
Y
.
In
a
similar
vein,
it
follows
immediately
from
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud],
that,
after
pos-
sibly
replacing
K
by
a
suitable
finite
extension
field
of
K
and
replacing
Y
†
by
a
suitable
Gal(Y
/X)-stable
dense
open
subscheme
of
Y
†
,
we
may
assume
without
loss
of
generality
that
Y
†
is
stabilized
by
the
action
of
Gal(Y
/X).
Write
f
†
:
Y
†
→
X
for
the
natural
dominant
morphism
that
restricts
to
the
finite
étale
Galois
cov-
ering
f
η
:
Y
→
X;
κ
†
:
Y
†
→
Y
st
for
the
natural
dominant
morphism
to
a
compactified
stable
model
Y
st
of
Y
over
O
K
[cf.
[ExtFam],
Theorem
A].
Thus,
for
each
positive
integer
i
≤
N
,
f
†
(D
i
†
)
=
x
i
∈
X
s
.
In
particular,
since
the
covering
f
η
:
Y
→
X
is
Galois,
and
Y
†
is
stabilized
by
the
action
of
Gal(Y
/X),
it
follows
immediately
from
Zariski’s
Main
Theorem
that,
for
each
positive
integer
i
≤
N
,
the
inverse
image
(f
†
)
−1
(x
i
)
⊆
Y
s
†
is
a
closed
subscheme
that
contains
D
i
†
and
is
pure
of
dimension
1.
Here,
we
recall
that
D
i
†
is
an
irreducible
component
of
Y
s
†
whose
normalization
is
of
genus
≥
1,
hence
necessarily
maps
birationally,
via
κ
†
,
to
an
irreducible
component
of
Y
st
.
In
particular,
we
conclude
that
each
connected
component
of
(f
†
)
−1
(x
i
)
⊆
Y
s
†
contains
an
irreducible
component
of
Y
s
†
that
maps
birationally,
via
κ
†
,
to
an
irreducible
component
of
Y
st
.
Next,
let
D
†
⊆
Y
s
†
be
an
irreducible
component
of
Y
s
†
that
maps
to
a
closed
point
κ
†
(D
†
)
of
Y
s
st
via
κ
†
:
Y
†
→
Y
st
,
but
dominates
an
irreducible
component
def
E
=
f
†
(D
†
)
of
X
s
.
Note
that
these
assumptions
imply
that
the
normalization
of
D
†
is
of
genus
0,
and
hence
that
E
is
an
irreducible
component
of
X
s
whose
normalization
is
of
genus
0.
Thus,
we
conclude
[cf.
the
condition
imposed
on
the
subset
{x
1
,
.
.
.
,
x
N
}
⊆
X
s
]
that
E
contains
three
points
∈
{x
1
,
.
.
.
,
x
N
},
i.e.,
[since
(f
†
)
−1
(x
i
)
is
pure
of
dimension
1]
that
D
†
contains
at
least
3
nodes
[that
map
to
three
distinct
“x
i
”].
On
the
other
hand,
this
[together
with
the
birationality
of
κ
†
]
implies
that
the
closed
point
κ
†
(D
†
)
of
Y
s
st
intersects
three
distinct
irreducible
components
of
Y
s
st
[i.e.,
the
images
of
suitable
irreducible
components
of
(f
†
)
−1
(x
i
)
⊆
Y
s
†
,
for
three
distinct
“i”],
that
is
to
say,
in
con-
tradiction
to
the
definition
of
the
notion
of
a
compactified
stable
model
[cf.
Definition
2.1,
(iv)].
Thus,
we
conclude
that
there
do
not
exist
any
such
“D
†
”
[i.e.,
that
map
to
a
closed
point
of
Y
s
st
,
but
dominate
an
irreducible
component
of
X
s
],
and
hence,
by
Zariski’s
Main
Theorem,
that
the
morphism
f
†
:
Y
†
→
X
factors
as
the
composite
of
κ
†
with
a
morphism
f
st
:
Y
st
→
X
.
In
particular,
it
follows
from
the
existence
of
the
morphism
f
st
:
Y
st
→
X
that
Y
s
st
contains
an
irreducible
component
whose
corresponding
valuation
induces
the
given
val-
uation
v
on
K(X),
i.e.,
that
X
satisfies
Σ-RNS.
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(iv).
In
light
of
Proposition
2.3,
(iii),
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
we
may
assume
without
loss
53
of
generality
that
X
is
a
compactified
semistable
model
with
split
reduction
of
X
over
O
K
.
Write
{v
1
,
.
.
.
,
v
N
}
for
the
set
of
discrete
valuations
on
K(X)
that
arise
from
the
irreducible
components
of
X
s
.
Then
since
X
satisfies
Σ-RNS,
for
each
positive
integer
i
≤
N
,
there
exists
a
connected
geometrically
pro-Σ
finite
étale
Galois
covering
Y
i
→
X
such
that
v
i
coincides
with
the
restriction
of
a
discrete
valuation
on
the
function
field
of
Y
i
that
arises
from
an
irreducible
component
of
the
special
fiber
of
a
compactified
stable
model
of
Y
i
.
Write
Y
→
X
for
the
composite
covering
of
the
connected
geometrically
pro-Σ
finite
étale
Galois
coverings
{Y
i
→
X}
1≤i≤N
.
Then
it
follows
immediately
from
Zariski’s
Main
Theorem
[cf.
also
[ExtFam],
Theorem
A]
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
there
exists
a
compactified
stable
model
Y
of
Y
over
O
K
that
dominates
X
.
This
completes
the
proof
of
assertion
(iv).
Assertion
(v)
and
the
necessity
portion
of
assertion
(vi)
follow
immediately
from
Proposition
2.4,
(iv),
together
with
Proposition
2.3,
(iv),
(vi).
Next,
we
be
an
element
consider
the
sufficiency
portion
of
assertion
(vi).
Let
c
∈
VE(
X)
that
extends
that
corresponds
[cf.
Proposition
2.3,
(v)]
to
a
valuation
v
on
K(
X)
a
discrete
residue-transcendental
p-valuation
on
X.
[Note
that
in
this
situation,
v
itself
is
necessarily
residue-transcendental.]
Then
it
suffices
to
show
that
there
exists
a
“z
c
”
as
in
the
statement
of
Proposition
2.4,
(vi),
that
is
a
generic
point
of
“Z
s
st
”.
To
this
end,
we
observe
that
the
nonexistence
of
such
a
“z
c
”
would
imply
that
all
of
the
“z
c
”
are
closed
points
of
“Z
s
st
”,
hence
have
residue
fields
that
are
algebraic
over
the
residue
field
of
O
K
.
On
the
other
hand,
this
would
imply
that
the
residue
field
of
R
c
=
O
v
is
algebraic
over
the
residue
field
of
O
K
,
in
contradiction
to
the
residue-transcendentality
of
v.
This
completes
the
proof
of
assertion
(vi).
Finally,
we
verify
assertion
(vii).
The
portion
of
assertion
(vii)
concerning
prim
as
in
the
statement
of
Proposition
the
existence
of
an
element
c
∈
VE(
X)
2.4,
(vii),
follows
from
Proposition
2.3,
(xi).
To
verify
the
portion
of
assertion
prim
,
it
suffices
(vii)
concerning
the
uniqueness
of
such
an
element
c
∈
VE(
X)
prim
that
satisfy
the
to
show
the
equality
of
any
two
elements
c
1
,
c
2
∈
VE(
X)
condition
imposed
on
element
“c”
in
the
statement
of
Proposition
2.4,
(vii).
st,prim
.
Then
it
follows
from
Write
c
1
,
c
2
for
the
images
of
c
1
,
c
2
in
VE(
X)
Proposition
2.4,
(iv);
[CbTpIV],
Corollary
1.15,
(iv)
[applied
to
c
1
,
c
2
],
that
δ(c
1
,
c
2
)
<
+∞,
hence
from
Proposition
2.3,
(ix),
that
c
1
=
c
2
,
as
desired.
This
completes
the
proof
of
assertion
(vii),
hence
of
Proposition
2.4.
Corollary
2.5
(Constructions
associated
to
geometric
tempered
fun-
damental
groups).
Let
Σ
⊆
Primes
be
a
subset
of
cardinality
≥
2
such
that
p
∈
Σ;
K
†
,
K
‡
mixed
characteristic
complete
discrete
valuation
fields
of
residue
†
‡
characteristic
p;
X
†
,
X
‡
hyperbolic
curves
over
K
,
K
,
respectively.
Write
Ω
†
,
†
‡
Ω
‡
for
the
p-adic
completions
of
K
,
K
,
respectively.
For
any
hyperbolic
curve
†
‡
Z
over
K
,
K
,
Ω
†
,
or
Ω
‡
,
write
Π
tp
Z
for
the
Σ-tempered
fundamental
group
of
Z,
relative
to
a
suitable
choice
of
basepoint
[cf.
the
subsection
in
Notations
54
†
→
X
†
,
X
‡
→
X
‡
and
Conventions
entitled
“Fundamental
groups”].
Write
X
tp
for
the
universal
geometrically
pro-Σ
coverings
corresponding
to
Π
X
†
,
Π
tp
,
re-
X
‡
spectively.
Suppose
that
X
†
and
X
‡
satisfy
Σ-RNS.
Then
the
following
hold
[cf.
Remark
2.5.1
below]:
∼
(i)
Let
σ
:
Π
tp
→
Π
tp
be
an
isomorphism
of
topological
groups.
Then
σ
X
†
X
‡
induces
homeomorphisms
∼
‡
),
†
)
→
VE(
X
VE(
X
∼
†
)
tor
→
‡
)
tor
,
VE(
X
VE(
X
∼
†
)
prim
→
VE(
X
‡
)
prim
,
VE(
X
that
are
compatible
with
the
respective
natural
actions
of
Π
tp
,
Π
tp
.
If,
X
†
X
‡
†
‡
moreover,
X
and
X
are
proper,
then
σ
induces
a
homeomorphism
∼
‡
)
an
†
)
an
→
(
X
(
X
that
is
compatible
with
the
respective
natural
actions
of
Π
tp
,
Π
tp
.
X
†
X
‡
†
def
‡
def
(ii)
Suppose
that
K
=
K
†
=
K
‡
,
K
=
K
=
K
,
hence
that
Ω
=
Ω
†
=
Ω
‡
.
Let
x
†
∈
X
†
(Ω),
x
‡
∈
X
‡
(Ω).
Write
X
x
†
†
(respectively,
X
x
‡
‡
)
for
the
∼
:
Π
tp
†
→
hyperbolic
curve
X
Ω
†
\{x
†
}
(respectively,
X
Ω
‡
\{x
‡
})
over
Ω.
Let
σ
X
†
x
tp
Π
‡
be
an
isomorphism
of
topological
groups
that
fits
into
a
commutative
X
‡
x
diagram
∼
Π
tp
†
−−−−→
Π
tp
‡
X
†
X
‡
σ
⏐
x
⏐
x
⏐
⏐
∼
Π
tp
−−−−→
Π
tp
,
X
†
X
‡
σ
where
the
vertical
arrows
are
the
natural
surjections
[determined
up
to
composition
with
an
inner
automorphism]
induced
by
the
natural
open
immersions
X
x
†
†
→
X
Ω
†
,
X
x
‡
‡
→
X
Ω
‡
of
hyperbolic
curves;
the
lower
hori-
zontal
arrow
σ
is
the
isomorphism
of
topological
groups
[determined
up
to
composition
with
an
inner
automorphism]
induced
by
a(n)
[uniquely
de-
∼
termined
—
cf.,
e.g.,
[DM],
Lemma
1.14]
isomorphism
σ
X
:
X
†
→
X
‡
of
schemes
over
K.
Then
x
‡
=
σ
X
(x
†
).
Proof.
First,
we
verify
assertion
(i).
We
begin
by
recalling
that
[SemiAn],
Corol-
lary
3.11,
may
be
generalized/applied
to
hyperbolic
curves
over
an
arbitrary
mixed
characteristic
complete
discrete
valuation
field
[cf.
[AbsTopII],
Remark
2.11.1,
(i)].
Thus,
by
applying
this
generalized
version
of
[SemiAn],
Corollary
∼
‡
)
st
.
†
)
st
→
VE(
X
3.11,
we
conclude
that
σ
induces
a
homeomorphism
VE(
X
†
‡
On
the
other
hand,
it
follows
from
our
assumption
that
X
and
X
satisfy
∼
→
st
”
and
Σ-RNS
that
we
may
apply
the
homeomorphisms
“VE(
X)
VE(
X)
55
∼
tor
→
st,tor
”
of
Proposition
2.4,
(v).
In
particular,
we
conclude
“VE(
X)
VE(
X)
that
σ
induces
a
homeomorphism
∼
‡
)
tor
†
)
tor
→
VE(
X
VE(
X
that
is
manifestly
compatible
with
the
respective
natural
actions
of
Π
tp
,
Π
tp
,
X
†
X
‡
as
well
as
a
homeomorphism
∼
‡
)
†
)
→
VE(
X
VE(
X
,
Π
tp
that
is
manifestly
compatible
with
the
respective
natural
actions
of
Π
tp
X
†
X
‡
and
preserves
specialization/generization
relations,
hence
induces
a
homeomor-
phism
∼
†
)
prim
→
‡
)
prim
VE(
X
VE(
X
that
is
compatible
with
the
respective
natural
actions
of
Π
tp
,
Π
tp
.
Finally,
if,
X
†
X
‡
†
‡
moreover,
X
and
X
are
proper,
then
we
may
apply
the
homeomorphism
“θ
X
”
of
Proposition
2.3,
(viii),
to
conclude
that
σ
induces
a
homeomorphism
∼
†
)
an
→
‡
)
an
(
X
(
X
that
is
compatible
with
the
respective
natural
actions
of
Π
tp
,
Π
tp
.
This
com-
X
†
X
‡
pletes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
We
begin
by
observing
that
it
follows
from
the
generalized
version
of
[SemiAn],
Corollary
3.11,
discussed
above,
together
with
Corollary
2.5,
(i),
that
σ
induces
a
bijection
∼
†
)
→
‡
)
VE(
X
VE(
X
that
maps
the
point-theoretic
orbit-center-chain
associated
to
x
†
to
the
point-
theoretic
orbit-center-chain
associated
to
x
‡
.
Since
σ
arises
from
σ
X
,
we
thus
∼
pt-th
/Gal(
X/X
conclude
from
the
bijection
“X(Ω)
→
VE(
X)
K
)”
of
Proposition
†
‡
2.3,
(v),
that
σ
X
(x
)
=
x
.
This
completes
the
proof
of
assertion
(ii),
hence
of
Corollary
2.5.
∼
†
)
an
→
‡
)
an
”
of
Corollary
2.5,
(i),
Remark
2.5.1.
The
homeomorphism
“(
X
(
X
is
essentially
similar
to
the
homeomorphisms
of
[Lpg1],
Theorem
3.10,
but
is
formulated
and
proven
according
to
the
approach
of
the
present
paper.
On
the
other
hand,
Corollary
2.5,
(ii),
may
be
regarded,
when
taken
together
with
Theorem
2.17
below,
as
a
generalization
of
[Tsjm],
Theorem
2.2;
its
proof
may
be
regarded
as
a
more
sophisticated
version
of
the
argument
applied
in
the
proof
of
[Tsjm],
Theorem
2.2.
Proposition
2.6
(Existence
of
new
ordinary
parts
of
certain
cover-
ings
after
Raynaud-Tamagawa).
Let
l
be
a
prime
number
=
p;
K
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
56
proper
hyperbolic
curve
over
K.
Suppose
that
X
has
split
stable
reduction
over
K.
Write
X
st
for
the
[unique,
up
to
unique
isomorphism]
stable
model
of
X
over
O
K
.
Suppose,
moreover,
that
every
irreducible
component
of
the
special
fiber
of
X
st
is
a
smooth
curve
of
genus
≥
2.
Write
e
X
(respectively,
v
X
)
for
the
cardinality
of
the
set
of
nodes
(respectively,
the
set
of
irreducible
components)
of
the
stable
curve
X
s
st
.
Then:
(i)
For
each
sufficiently
large
positive
integer
m,
if
we
replace
K
by
a
finite
unramified
extension
field
of
K,
then
there
exists
a
finite
étale
cyclic
cov-
ering
Y
st
−→
X
st
over
O
K
of
degree
l
m
satisfying
the
following
conditions:
(a)
Write
(Y
st
→)
Z
st
→
X
st
for
the
finite
étale
cyclic
subcovering
over
O
K
of
degree
l
m−1
;
Y
,
Z
for
the
generic
fibers
of
Y
st
,
Z
st
,
respec-
tively.
Then
Y
and
Z
have
split
stable
reduction
over
K.
Moreover,
Y
st
,
Z
st
are
the
stable
models
of
Y
,
Z,
respectively.
(b)
The
finite
étale
covering
Y
s
st
→
X
s
st
determined
by
the
finite
étale
cyclic
covering
Y
st
→
X
st
induces
a
bijection
between
the
respective
sets
of
irreducible
components.
(c)
Write
A
for
the
abelian
variety
over
K
obtained
by
forming
the
coker-
nel
of
the
natural
morphism
J(Z)
→
J(Y
)
induced
by
the
finite
étale
cyclic
covering
Y
→
Z
[of
degree
l].
Then
there
exists
an
abelian
va-
riety
B
over
K
with
good
ordinary
reduction
such
that
T
p
A
fits
into
exact
sequences
of
G
K
-modules
[cf.
the
theory
of
[FC],
especially,
[FC],
Chapter
III,
Corollary
7.3]
0
−→
T
gd
−→
T
p
A
−→
T
cb
−→
0
0
−→
T
tor
−→
T
gd
−→
T
p
B
−→
0
0
−→
Hom(T
p
B
s
,
Z
p
(1))
−→
T
p
B
−→
T
p
B
s
−→
0,
where
“(1)”
denotes
the
Tate
twist;
the
natural
action
of
G
K
on
the
“combinatorial
quotient”
T
cb
[i.e.,
the
inverse
limit
of
the
quotients
“Y
/nY
”
of
[FC],
Chapter
III,
Corollary
7.3,
as
n
ranges
over
the
positive
integral
powers
of
p]
of
T
p
A
is
trivial;
T
tor
is
isomorphic
as
a
G
K
-module
to
the
direct
sum
of
a
collection
of
copies
of
Z
p
(1);
B
denotes
the
abelian
scheme
over
O
K
whose
generic
fiber
is
equal
to
B.
(ii)
Fix
a
finite
étale
cyclic
covering
Y
st
→
X
st
as
in
(i).
Write
T
cb,Y
,
T
cb,Z
for
the
“combinatorial
quotients”
[i.e.,
the
inverse
limit
of
the
quotients
“Y
/nY
”
of
[FC],
Chapter
III,
Corollary
7.3,
as
n
ranges
over
the
positive
integral
powers
of
p]
of
T
p
J(Y
),
T
p
J(Z),
respectively;
h
Y
,
h
Z
for
the
57
respective
loop-ranks
of
the
dual
graphs
associated
to
the
stable
curves
Y
s
st
,
Z
s
st
.
Then
it
holds
that
h
Y
=
1
+
l
m
e
X
−
v
X
,
h
Z
=
1
+
l
m−1
e
X
−
v
X
.
Moreover,
rank
Z
p
T
cb,Y
=
h
Y
;
rank
Z
p
T
cb,Z
=
h
Z
;
rank
Z
p
T
tor
=
rank
Z
p
T
cb
=
h
Y
−
h
Z
=
(l
m
−
l
m−1
)e
X
.
Proof.
First,
we
verify
assertion
(i).
Write
{C
i
}
1≤i≤v
X
for
the
set
of
irreducible
components
of
X
s
st
.
Let
m
be
a
positive
integer
such
that,
for
each
positive
integer
i
≤
v
X
,
it
holds
that
l
m
>
l
2g
Ci
−
l
2g
Ci
−1
(p
−
1)g
C
i
,
l
2g
Ci
−
1
where
g
(−)
denotes
the
genus
of
(−).
Then,
in
light
of
[Tama1],
Lemma
1.9
[i.e.,
a
generalization
of
[Ray1],
Théorème
4.3.1],
by
replacing
K
by
a
finite
unramified
extension
field
of
K,
one
may
construct
finite
étale
cyclic
coverings
{D
i
→
C
i
}
1≤i≤v
X
of
degree
l
m
[of
proper
hyperbolic
curves
over
the
residue
field
of
K]
satisfying
the
following
conditions:
•
For
each
positive
integer
i
≤
v
X
,
write
(D
i
→)
E
i
→
C
i
for
the
finite
étale
cyclic
subcovering
of
degree
l
m−1
[of
proper
hyperbolic
curves
over
the
residue
field
of
K].
Then
the
abelian
variety
obtained
by
forming
the
cokernel
of
the
natural
morphism
J(E
i
)
→
J(D
i
)
induced
by
the
finite
étale
cyclic
covering
D
i
→
E
i
of
degree
l
is
ordinary.
•
The
cardinality
of
the
set
of
closed
points
of
D
i
that
lie
over
the
closed
points
of
C
i
determined
by
the
nodes
of
X
s
st
is
equal
to
l
m
.
Next,
one
verifies
immediately
that
there
exists
a
finite
étale
cyclic
covering
D
→
X
s
st
of
degree
l
m
obtained
by
gluing
together
the
finite
étale
cyclic
cov-
erings
{D
i
→
C
i
}
1≤i≤v
X
.
Write
Y
st
→
X
st
for
the
finite
étale
cyclic
covering
obtained
by
deforming
the
finite
étale
cyclic
covering
D
→
X
s
st
.
Then
it
follows
immediately
from
the
various
definitions
involved
that
conditions
(a),
(b)
hold.
Moreover,
in
light
of
the
theory
of
Raynaud
extensions
[cf.
[FC],
Chapter
II,
§1;
[FC],
Chapter
III,
Corollary
7.3],
together
with
Remark
2.6.1,
(i),
(ii),
below
[cf.
also
[BLR],
§9.2,
Example
8],
one
concludes
that
condition
(c)
holds.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Write
e
Y
,
e
Z
for
the
respective
cardinalities
of
the
sets
of
nodes
of
the
stable
curves
Y
s
st
,
Z
s
st
;
v
Y
,
v
Z
for
the
respective
cardinalities
of
the
sets
of
irreducible
components
of
the
stable
curves
Y
s
st
,
Z
s
st
.
Then
it
follows
immediately
from
conditions
(a),
(b),
that
e
Y
=
l
m
e
X
,
e
Z
=
l
m−1
e
X
,
58
v
Y
=
v
Z
=
v
X
.
Thus,
we
conclude
from
the
well-known
computation
of
the
first
homology
group
of
a
finite
graph
that
h
Y
=
1
+
e
Y
−
v
Y
=
1
+
l
m
e
X
−
v
X
,
h
Z
=
1
+
e
Z
−
v
Z
=
1
+
l
m−1
e
X
−
v
X
.
Therefore,
to
complete
the
proof
of
assertion
(ii),
it
suffices
to
prove
that
rank
Z
p
T
cb,Y
=
h
Y
,
rank
Z
p
T
cb,Z
=
h
Z
,
and
rank
Z
p
T
tor
=
rank
Z
p
T
cb
=
h
Y
−h
Z
.
Recall
that
the
loop-ranks
h
Y
,
h
Z
coincide
with
the
toric
ranks
of
the
Jacobians
of
the
stable
curves
Y
s
st
,
Z
s
st
,
respectively
[cf.,
e.g,
[BLR],
§9.2,
Example
8].
On
the
other
hand,
in
light
of
the
theory
of
duality
for
torsion
subgroups
of
abelian
varieties,
it
holds
that
these
toric
ranks
coincide
with
the
ranks
of
the
respective
corresponding
combinatorial
quotients
[cf.
[FC],
Chapter
III,
Corollary
7.4].
In
particular,
it
follows
immediately
[cf.
Remark
2.6.1,
(ii),
below;
[FC],
Chap-
ter
III,
Corollary
7.4]
that
rank
Z
p
T
cb,Y
=
h
Y
,
rank
Z
p
T
cb,Z
=
h
Z
,
hence
that
rank
Z
p
T
tor
=
rank
Z
p
T
cb
=
rank
Z
p
T
cb,Y
−
rank
Z
p
T
cb,Z
=
h
Y
−
h
Z
.
This
completes
the
proof
of
assertion
(ii),
hence
of
Proposition
2.6.
Remark
2.6.1.
We
maintain
the
notation
of
Proposition
2.6.
(i)
Write
A
for
the
identity
component
of
the
Néron
model
of
A
over
O
K
[cf.
[BLR],
§1.3,
Corollary
2].
Then
the
universal
property
of
the
Néron
model
implies
the
existence
of
a
surjective
homomorphism
f
:
Pic
0
Y
st
/O
K
A
that
extends
the
natural
quotient
homomorphism
J(Y
)
A
[cf.
[BLR],
§1.2,
Definition
1;
[BLR],
§9.4,
Theorem
1].
Thus,
since
Pic
0
Y
st
/O
K
is
a
semi-abelian
scheme
over
O
K
[cf.
[BLR],
§9.4,
Theorem
1],
it
follows
immediately
from
the
existence
of
the
surjective
homomorphism
f
that
A
is
also
a
semi-abelian
scheme
over
O
K
.
(ii)
Recall
that
the
composite
homomorphism
J(Z)
→
J(Y
)
→
J(Z)
of
the
norm
map
J(Y
)
→
J(Z)
with
the
natural
homomorphism
J(Z)
→
J(Y
)
coincides
with
the
morphism
given
by
multiplication
by
l.
In
particular,
the
abelian
variety
J(Y
)
is
isogenous
over
K
to
the
product
abelian
variety
J(Z)
×
K
A.
Thus,
we
conclude
from
[BLR],
§7.3,
Proposition
6,
that
the
semi-abelian
schemes
Pic
0
Y
st
/O
K
and
Pic
0
Z
st
/O
K
×
O
K
A
over
O
K
[cf.
[BLR],
§9.4,
Theorem
1]
are
isogenous
over
O
K
.
Definition
2.7.
In
the
notation
of
Remark
2.6.1,
let
Z
be
a
semistable
model
of
Z
over
O
K
that
has
split
reduction.
Note
that
Z
st
satisfies
this
property,
and
that,
by
pulling-back
the
finite
étale
cyclic
covering
Y
st
→
Z
st
via
the
unique
morphism
Z
→
Z
st
that
extends
the
identity
morphism
Z
→
Z
[cf.
[ExtFam],
Theorem
A],
we
obtain
a
finite
étale
cyclic
covering
Y
−→
Z
over
O
K
of
degree
l
that
extends
the
finite
étale
cyclic
covering
Y
→
Z
over
K.
Suppose
that
59
•
X
s
st
is
a
singular
curve,
and
that
•
m
is
sufficiently
large
that
h
Y
≥
h
Z
≥
1
[cf.
Proposition
2.6,
(ii)].
(i)
Let
y
η
∈
Y
(K).
Write
z
η
∈
Z(K)
for
the
image
of
y
η
via
the
natural
map
Y
(K)
→
Z(K).
Then
y
η
and
z
η
determine
embeddings
Y
→
J(Y
),
Z
→
J(Z)
that
allow
one
to
regard
J(Y
),
J(Z)
as
the
respective
Albanese
varieties
of
Y
,
Z
[cf.
[AbsTopI],
Appendix,
Definition
A.1,
(ii);
[Milne],
Proposition
6.1].
In
particular,
we
obtain
a
commutative
diagram
Δ
Y
−−−−→
T
p
J(Y
)
−−−−→
T
cb,Y
⏐
⏐
⏐
⏐
⏐
⏐
Δ
Z
−−−−→
T
p
J(Z)
−−−−→
T
cb,Z
,
where
the
left-hand
vertical
arrow
denotes
the
open
injection
induced
by
the
finite
étale
cyclic
covering
Y
→
Z;
the
left-hand
horizontal
arrows
denote
the
natural
surjections
determined
by
the
Albanese
embeddings
Y
→
J(Y
),
Z
→
J(Z)
[cf.
[AbsTopI],
Appendix,
Proposition
A.6,
(iv)];
the
right-hand
horizontal
arrows
denote
the
natural
surjections;
the
mid-
dle
and
right-hand
vertical
arrows
are
surjections
[cf.
the
fact
that
the
finite
étale
cyclic
covering
Y
→
Z
is
of
degree
l
=
p];
the
first
square
of
the
diagram
commutes
in
light
of
the
functoriality
of
the
étale
funda-
mental
group;
the
second
square
of
the
diagram
commutes
in
light
of
the
functoriality
of
Raynaud
extensions.
(ii)
Fix
a
quotient
T
cb,Z
Z
p
[cf.
our
assumption
that
h
Z
≥
1;
Proposition
2.6,
(ii)].
For
each
nonneg-
ative
integer
n,
write
Z
n
−→
Z
for
the
finite
étale
cyclic
[“combinatorial”]
covering
of
degree
p
n
over
O
K
induced
by
the
natural
quotient
(Δ
Z
T
p
J(Z)
)
T
cb,Z
Z
p
Z/p
n
Z;
Y
n
−→
Y
for
the
finite
étale
cyclic
[“combinatorial”]
covering
of
degree
p
n
over
O
K
induced
by
the
natural
quotient
(Δ
Y
T
p
J(Y
)
)
T
cb,Y
T
cb,Z
Z
p
Z/p
n
Z.
Thus,
the
commutative
diagram
in
(i)
induces
a
cartesian
commutative
diagram
Y
n
−−−−→
Y
⏐
⏐
⏐
⏐
Z
n
−−−−→
Z,
60
where
the
vertical
arrows
are
finite
étale
cyclic
coverings
of
degree
l
over
O
K
;
the
horizontal
arrows
are
finite
étale
cyclic
[“combinatorial”]
cover-
ings
of
degree
p
n
over
O
K
.
Moreover:
(a)
Write
Y
n
,
Z
n
for
the
generic
fibers
of
Y
n
,
Z
n
,
respectively.
Then
the
finite
étale
covering
(Y
n
)
s
→
(Z
n
)
s
determined
by
the
finite
étale
cyclic
covering
Y
n
→
Z
n
induces
a
bijection
between
the
sets
of
irreducible
components
that
arise
from
the
respective
stable
models
of
Y
n
and
Z
n
[cf.
Proposition
2.6,
(i),
(b)].
(b)
Write
A
n
for
the
abelian
variety
over
K
obtained
by
forming
the
cokernel
of
the
natural
morphism
J(Z
n
)
→
J(Y
n
)
induced
by
the
finite
étale
cyclic
covering
Y
n
→
Z
n
of
degree
l
[of
proper
hyperbolic
curves
over
K].
Then
there
exists
an
abelian
variety
B
n
over
K
with
good
ordinary
reduction
such
that
T
p
A
n
fits
into
exact
sequences
of
G
K
-modules
[cf.
the
theory
of
[FC],
especially,
[FC],
Chapter
III,
Corollary
7.3;
the
proof
of
Proposition
2.6,
(i),
(c)]
0
−→
T
gd,n
−→
T
p
A
n
−→
T
cb,n
−→
0
0
−→
T
tor,n
−→
T
gd,n
−→
T
p
B
n
−→
0
0
−→
Hom(T
p
(B
n
)
s
,
Z
p
(1))
−→
T
p
B
n
−→
T
p
(B
n
)
s
−→
0,
where
“(1)”
denotes
the
Tate
twist;
the
natural
action
of
G
K
on
the
“combinatorial
quotient”
T
cb,n
of
T
p
A
n
is
trivial;
T
tor,n
is
isomorphic
as
a
G
K
-module
to
the
direct
sum
of
a
collection
of
copies
of
Z
p
(1);
B
n
denotes
the
abelian
scheme
over
O
K
whose
generic
fiber
is
equal
to
B
n
.
(iii)
Write
A
n
for
the
identity
component
of
the
Néron
model
of
A
n
over
O
K
[cf.
[BLR],
§1.3,
Corollary
2].
Then
the
universal
property
of
the
Néron
model
implies
the
existence
of
a
surjective
homomorphism
def
f
n
:
J
n
=
Pic
0
Y
n
/O
K
A
n
that
extends
the
natural
quotient
homomorphism
J(Y
n
)
A
n
[cf.
[BLR],
§1.2,
Definition
1;
[BLR],
§9.4,
Theorem
1].
Thus,
since
J
n
=
Pic
0
Y
n
/O
K
is
a
semi-abelian
scheme
over
O
K
[cf.
[BLR],
§9.4,
Theorem
1],
it
follows
immediately
from
the
existence
of
the
surjective
homomorphism
f
n
that
A
n
is
also
a
semi-abelian
scheme
over
O
K
[cf.
Remark
2.6.1,
(i)].
(iv)
For
each
nonnegative
integer
n,
write
h
Y
n
,
61
h
Z
n
for
the
loop-ranks
of
the
dual
graphs
associated
to
the
semistable
curves
(Y
n
)
s
,
(Z
n
)
s
,
respectively;
g
Y
n
for
the
[arithmetic]
genus
of
the
semistable
model
Y
n
over
O
K
.
Then
a
similar
argument
to
the
argument
applied
in
Remark
2.6.1,
(ii),
implies
that
the
semi-abelian
schemes
Pic
0
Y
n
/O
K
and
Pic
0
Z
n
/O
K
×
O
K
A
n
over
O
K
are
isogenous
over
O
K
.
In
particular,
we
obtain
equalities
rank
Z
p
T
tor,n
=
rank
Z
p
T
cb,n
=
h
Y
n
−
h
Z
n
[cf.
the
proof
of
Proposition
2.6,
(ii)].
Proposition
2.8
(Explicit
computations
of
toric
rank
and
genus).
We
maintain
the
notation
of
Definition
2.7.
Then
the
following
hold:
(i)
It
holds
that
h
Y
n
=
1
+
p
n
l
m
e
X
−
p
n
v
X
,
h
Z
n
=
1
+
p
n
l
m−1
e
X
−
p
n
v
X
,
hence,
in
particular,
that
rank
Z
p
T
tor,n
=
rank
Z
p
T
cb,n
=
h
Y
n
−
h
Z
n
=
p
n
(l
m
−
l
m−1
)e
X
[cf.
the
final
display
of
Definition
2.7,
(iv)].
(ii)
It
holds
that
g
Y
n
=
p
n
(g
Y
0
−
1)
+
1.
Proof.
First,
recall
from
the
well-known
theory
of
stable
and
semistable
models
that
h
Y
n
,
h
Z
n
,
rank
Z
p
T
tor,n
,
rank
Z
p
T
cb,n
,
and
g
Y
n
are
independent
of
the
choice
of
the
semistable
model
Z
of
Z
over
O
K
.
[Indeed,
by
passing
to
a
suitable
finite
unramified
extension
of
K
and
adding
finitely
many
suitably
positioned
cusps,
one
may,
in
effect,
reduce
this
“well-known
theory
of
stable
and
semistable
models”
to
the
theory
of
pointed
stable
curves
and
contraction
morphisms
that
arise
from
eliminating
cusps,
as
exposed
in
[Knud].]
In
particular,
we
may
assume
without
loss
of
generality
that
Z
is
the
stable
model
of
Z
over
O
K
.
Assertion
(i)
then
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
Proposition
2.6,
(ii).
Assertion
(ii)
follows
immediately
from
Hurwitz’s
formula.
This
completes
the
proof
of
Proposition
2.8.
Definition
2.9.
We
maintain
the
notation
of
Definition
2.7.
Then
we
shall
write
T
cnn,n
62
for
the
connected
p-divisible
group
over
O
K
that
arises
as
the
connected
part
of
the
p-divisible
group
[cf.
the
discussion
preceding
[Tate],
§2.2,
Proposition
2]
associated
to
the
Raynaud
extension
[cf.
[FC],
Chapter
II,
§1]
of
A
n
;
T
tor,n
for
the
connected
p-divisible
group
over
O
K
associated
to
the
torus
that
appears
in
the
Raynaud
extension
of
A
n
;
T
cnn,n
,
T
tor,n
for
the
respective
generic
fibers
of
T
cnn,n
,
T
tor,n
;
def
T
cnn,n
=
T
p
(
T
cnn,n
),
def
T
tor,n
=
T
p
(
T
tor,n
)
for
the
respective
p-adic
Tate
modules
of
T
cnn,n
,
T
tor,n
[cf.
the
subsection
in
Notations
and
Conventions
entitled
“Schemes”].
Note
that
T
cnn,n
and
T
tor,n
may
be
regarded
as
Z
p
-submodules
of
T
p
A
n
in
a
natural
way.
Moreover,
we
shall
write
def
T
ét,n
=
T
p
A
n
/T
cnn,n
.
Note
that
G
K
acts
naturally
on
the
Z
p
-modules
T
cnn,n
,
T
tor,n
,
T
ét,n
,
and
T
cb,n
[cf.
Definition
2.7,
(ii),
(b)].
We
shall
write
T
cnn,n
T
qtr,n
for
the
maximal
torsion-free
G
K
-stable
quotient
Z
p
-module
among
the
torsion-
free
G
K
-stable
quotient
Z
p
-modules
T
cnn,n
T
such
that
some
open
subgroup
of
G
K
acts
on
T
via
the
p-adic
cyclotomic
character;
T
qcb,n
⊆
T
ét,n
for
the
maximal
G
K
-stable
Z
p
-submodule
among
the
G
K
-stable
Z
p
-submodules
T
⊆
T
ét,n
such
that
some
open
subgroup
of
G
K
acts
trivially
on
T
.
Finally,
we
observe
that
[one
verifies
immediately
that]
we
obtain
natural
exact
sequences
of
G
K
-modules
[cf.
Definition
2.7,
(ii),
(b)]
0
−→
T
tor,n
−→
T
cnn,n
−→
Hom(T
p
(B
n
)
s
,
Z
p
(1))
−→
0
0
−→
T
p
(B
n
)
s
−→
T
ét,n
−→
T
cb,n
−→
0
0
−→
T
cnn,n
−→
T
p
A
n
−→
T
ét,n
−→
0.
Lemma
2.10.
We
maintain
the
notation
of
Definition
2.9.
Then:
(i)
The
natural
action
of
G
K
on
T
p
A
n
induces
the
trivial
action
of
I
K
on
T
ét,n
.
63
(ii)
There
exist
natural
compatible
G
K
-equivariant
isomorphisms
∼
T
ét,n
→
Hom(T
cnn,n
,
Z
p
(1)),
∼
T
cb,n
→
Hom(T
tor,n
,
Z
p
(1)),
∼
T
qcb,n
→
Hom(T
qtr,n
,
Z
p
(1)).
(iii)
Suppose
that
K
is
a
p-adic
local
field.
Then
the
set
of
eigenvalues
of
the
Z
p
-linear
automorphism
of
T
p
(B
n
)
s
induced
by
the
Frobenius
element
of
G
K
/I
K
[cf.
(i);
the
second
exact
sequence
of
Definition
2.9]
does
not
contain
any
roots
of
unity.
Proof.
First,
we
verify
assertion
(i).
Recall
that
the
quotient
of
a
p-divisible
group
by
its
connected
part
is
étale
[cf.,
e.g.,
the
discussion
preceding
[Tate],
§2.2,
Proposition
2].
Thus,
we
conclude
[cf.
the
triviality
of
the
action
of
G
K
on
T
cb,n
observed
in
Definition
2.7,
(ii),
(b);
the
second
exact
sequence
of
the
final
display
of
Definition
2.9]
from
[the
second
sentence
of]
[FC],
Chapter
III,
Corollary
7.3,
that
the
natural
action
of
I
K
on
T
ét,n
is
trivial,
as
desired.
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
the
theory
of
duality
for
torsion
subgroups
of
abelian
varieties
[cf.
[FC],
Chapter
III,
Corollary
7.4],
together
with
the
first
and
second
exact
sequences
of
the
final
display
of
Definition
2.9.
Assertion
(iii)
follows
immediately
from
the
finiteness
of
the
set
of
rational
points
of
(B
n
)
s
over
any
finite
extension
field
of
the
[finite!]
residue
field
of
K.
This
completes
the
proof
of
Lemma
2.10.
Proposition
2.11
(Toral
quotient
of
the
connected
part).
In
the
notation
of
Definition
2.9,
write
χ
n
:
T
cnn,n
T
qtr,n
for
the
natural
surjection
of
G
K
-modules.
Then
the
following
hold:
(i)
Suppose
that
the
residue
field
of
K
is
separably
closed.
Then
T
cnn,n
=
T
qtr,n
=
{0},
T
qcb,n
=
T
ét,n
=
{0}.
(ii)
Suppose
that
K
is
a
p-adic
local
field.
Then
the
restriction
of
χ
n
to
T
tor,n
⊆
T
cnn,n
induces
an
injection
T
tor,n
→
T
qtr,n
with
finite
cokernel.
Proof.
First,
we
verify
assertion
(i).
Note
that
since
the
residue
field
of
K
is
separably
closed,
G
K
=
I
K
.
Thus,
it
follows
immediately
from
Lemma
2.10,
(i),
together
with
the
various
definitions
involved,
that
T
qcb,n
=
T
ét,n
.
On
the
other
hand,
it
follows
immediately
from
Proposition
2.8,
(i)
[cf.
also
Proposition
2.6;
Definition
2.7;
the
second
exact
sequence
of
the
final
display
of
Definition
2.9],
that
T
ét,n
=
{0}.
Thus,
we
conclude
from
Lemma
2.10,
(ii),
that
T
cnn,n
=
T
qtr,n
=
{0}.
This
completes
the
proof
of
assertion
(i).
def
Next,
we
verify
assertion
(ii).
Write
N
=
rank
Z
p
T
qtr,n
.
Note
that,
in
light
of
the
maximality
of
T
qtr,n
,
it
suffices
to
verify
that
there
exists
a
unique
torsion-free
G
K
-stable
quotient
Z
p
-module
T
cnn,n
T
64
whose
restriction
to
T
tor,n
⊆
T
cnn,n
induces
an
injection
T
tor,n
→
T
with
finite
cokernel,
and
that
rank
Z
p
T
≥
N
.
Let
Q
p
be
an
algebraic
closure
of
Q
p
equipped
with
the
trivial
action
of
G
K
.
Then
observe
that
it
follows
from
a
routine
argument
involving
Galois
descent
from
Q
p
to
Q
p
that
it
suffices
to
verify
that
there
exists
a
unique
G
K
-stable
quotient
Q
p
-vector
space
T
cnn,n
⊗
Z
p
Q
p
−→
V
whose
restriction
to
T
tor,n
⊗
Z
p
Q
p
⊆
T
cnn,n
⊗
Z
p
Q
p
induces
an
isomorphism
∼
T
tor,n
⊗
Z
p
Q
p
→
V
,
and
that
dim
Q
p
V
≥
N
.
Write
k
for
the
finite
residue
field
of
the
p-adic
local
field
K
[so
G
K
/I
K
may
be
identified
with
the
absolute
Galois
group
G
k
of
k].
Then,
by
applying
Lemma
2.10,
(i),
(ii),
we
conclude
that
it
suffices
to
verify
that
there
exists
a
unique
G
k
-stable
Q
p
-subspace
V
∗
⊆
T
ét,n
⊗
Z
p
Q
p
whose
composite
with
the
natural
surjection
T
ét,n
⊗
Z
p
Q
p
T
cb,n
⊗
Z
p
Q
p
∼
induces
an
isomorphism
V
∗
→
T
cb,n
⊗
Z
p
Q
p
,
and
that
dim
Q
p
V
∗
≥
N
.
On
the
other
hand,
in
light
of
the
eigenspace
decomposition
associated
to
the
natural
action
of
the
Frobenius
element
∈
G
k
,
the
existence
and
uniqueness
of
such
a
subspace,
together
with
the
inequality
dim
Q
p
V
∗
≥
N
(=
rank
Z
p
T
qcb,n
)
[cf.
Lemma
2.10,
(ii)],
follows
immediately
from
Lemma
2.10,
(iii)
[cf.
also
the
triviality
of
the
action
of
G
K
on
T
cb,n
observed
in
Definition
2.7,
(ii),
(b)].
This
completes
the
proof
of
assertion
(ii),
hence
of
Proposition
2.11.
Proposition
2.12
(Construction
of
a
certain
morphism
of
formal
schemes
to
the
quasi-toral
quotient).
In
the
notation
of
Proposition
2.11,
let
y
n
∈
Y
n
(O
K
)
be
an
O
K
-rational
point
that
maps
the
closed
point
of
Spec
O
K
to
a
smooth
point
(y
n
)
s
of
the
semistable
curve
(Y
n
)
s
.
Write
y
n,η
∈
Y
n
(K)
for
the
K-valued
point
of
Y
n
determined
by
y
n
∈
Y
n
(O
K
);
C
⊆
(Y
n
)
s
for
the
unique
irreducible
component
that
contains
(y
n
)
s
;
F
⊆
Y
n
for
the
closed
subset
obtained
by
forming
the
union
of
the
irreducible
components
=
C
of
(Y
n
)
s
;
U
y
n
⊆
Y
n
for
the
open
subscheme
obtained
by
forming
the
complement
of
F
⊆
Y
n
;
h
n,η
:
Y
n
−→
J(Y
n
)
for
the
Albanese
map
that
maps
y
n,η
to
the
origin
[cf.
[AbsTopI],
Appendix,
Definition
A.1,
(ii);
[Milne],
Proposition
6.1].
Recall
that
J
n
is
a
semi-abelian
scheme
over
O
K
[cf.
Definition
2.7,
(iii)]
whose
generic
fiber
is
J(Y
n
).
In
particular,
J
n
is
isomorphic
to
the
identity
component
of
the
Néron
model
over
O
K
of
J(Y
n
)
[cf.
[BLR],
§7.4,
Proposition
3].
Thus,
since
U
y
n
is
a
connected
smooth
scheme
over
O
K
whose
generic
fiber
is
Y
n
,
the
universal
property
of
the
Néron
model
implies
the
existence
of
a
unique
morphism
h
n
:
U
y
n
−→
J
n
that
extends
h
n,η
.
Next,
write
J
n
,
A
n
for
the
formal
completions
at
the
origin
of
Y
,y
,
O
U
,y
for
the
completions
the
semi-abelian
schemes
J
n
,
A
n
over
O
K
;
O
n
n
yn
n
65
at
y
n
of
O
Y
n
,(y
n
)
s
,
O
U
yn
,(y
n
)
s
.
Then
the
natural
composite
map
Y
,y
=
Spf
O
U
,y
−→
J
n
−→
A
n
Spf
O
n
n
yn
n
induced
by
h
n
and
the
surjective
homomorphism
f
n
:
J
n
A
n
[cf.
Definition
2.7,
(iii)]
determines
a
morphism
of
formal
O
K
-schemes
Y
,y
−→
T
cnn,n
,
Spf
O
n
n
where
we
regard
the
connected
p-divisible
group
T
cnn,n
as
a
formal
group
over
O
K
[cf.
[Tate],
§2.2,
Proposition
1].
In
particular,
by
forming
the
composite
with
the
morphism
of
formal
O
K
-schemes
induced
by
χ
n
[cf.
[Tate],
§2.2,
Proposition
1;
[Tate],
§4.2,
Corollary
1],
we
obtain
a
morphism
of
formal
O
K
-schemes
Y
,y
−→
T
qtr,n
,
Spf
O
n
n
where
T
qtr,n
denotes
the
formal
group
over
O
K
determined
by
the
connected
p-
divisible
group
associated
to
the
G
K
-module
T
qtr,n
[cf.
[Tate],
§2.2,
Proposition
1;
[Tate],
§4.2,
Corollary
1].
Proof.
Proposition
2.12
follows
immediately
from
the
various
references
quoted
in
the
statement
of
Proposition
2.12.
Proposition
2.13
(Coverings
associated
to
characters).
We
maintain
the
notation
of
Proposition
2.12.
Then
the
following
hold:
m
for
the
formal
completion
at
the
origin
of
the
multiplicative
(i)
Write
G
m
)
for
the
Z
p
-module
of
group
scheme
G
m
over
O
K
;
Hom
O
K
(
T
qtr,n
,
G
homomorphisms
over
O
K
from
T
qtr,n
to
G
m
;
Hom
G
K
(T
qtr,n
,
Z
p
(1))
for
the
Z
p
-module
of
G
K
-equivariant
homomorphisms
of
Z
p
-modules
T
qtr,n
→
Z
p
(1).
Then
the
natural
homomorphism
m
)
−→
Hom
G
(T
qtr,n
,
Z
p
(1))
Hom
O
K
(
T
qtr,n
,
G
K
is
bijective.
m
).
Consider
the
com-
(ii)
Let
a
be
a
positive
integer;
f
∈
Hom
O
K
(
T
qtr,n
,
G
posite
∼
m
Y
,y
−→
G
(Spf
O
K
[[t]]
→)
Spf
O
n
n
[where
t
is
an
indeterminate,
and
we
regard
O
K
[[t]]
as
being
equipped
with
the
t-adic
topology]
of
the
morphism
in
the
final
display
of
Proposition
2.12
with
f
.
By
a
slight
abuse
of
notation,
we
shall
also
write
f
∈
O
K
[[t]]
×
m
via
the
homomorphism
for
the
image
of
the
canonical
coordinate
U
of
G
of
rings
induced
by
the
above
composite
morphism.
Then
the
covering
of
Spf
O
K
[[t]]
obtained
by
extracting
a
p
a
-th
root
of
f
is
dominated
by
the
covering
of
Spf
O
K
[[t]]
obtained
by
restricting
the
covering
determined
by
multiplication
by
p
a
on
A
n
.
66
Proof.
Assertion
(i)
follows
immediately
from
[Tate],
§2.2,
Proposition
1;
[Tate],
§4.2,
Corollary
1.
Assertion
(ii)
follows
immediately
from
the
various
definitions
involved.
This
completes
the
proof
of
Proposition
2.13.
Lemma
2.14.
Let
X
be
a
smooth
proper
curve
of
genus
g
X
over
a
field
K;
x
∈
X(K)
a
K-valued
point
of
X;
d
x
a
nonnegative
integer
such
that
d
x
≥
2g
X
−
1.
Then
the
natural
composite
map
H
0
(X,
Ω
X
)
→
Ω
X,x
Ω
X,x
/m
d
x
x
Ω
X,x
is
injective.
Proof.
First,
observe
that
since
d
x
≥
2g
X
−
1
>
2g
X
−
2
[which
implies
that
the
degree
of
the
line
bundle
Ω
X
(−d
x
·x)
is
negative],
it
follows
that
H
0
(X,
Ω
X
(−d
x
·
x))
=
0.
Thus,
the
desired
injectivity
follows
immediately
by
applying
the
[left
exact]
functor
H
0
(X,
−)
to
the
short
exact
sequence
0
−→
Ω
X
(−d
x
)
−→
Ω
X
−→
Ω
X,x
/m
d
x
x
Ω
X,x
−→
0.
This
completes
the
proof
of
Lemma
2.14.
Lemma
2.15.
Let
a,
b
be
positive
integers;
K
a
p-adic
local
field
of
degree
def
d
K
=
[K
:
Q
p
]
over
Q
p
;
W
⊆
K[T
]/(T
b+1
)
a
Q
p
-vector
subspace
of
dimension
a.
For
each
nonzero
element
h
i
T
i
∈
K[T
]/(T
b+1
),
h
=
0≤i≤b
def
write
ord(h)
(≤
b)
for
the
smallest
integer
i
such
that
h
i
=
0.
Set
ord(0)
=
+∞.
Suppose
that
def
W
\{0}
⊆
F
=
{h
∈
K[T
]/(T
b+1
)
|
ord(h)
=
p
j
−1
for
some
nonnegative
integer
j}.
Then
it
holds
that
a
≤
d
K
(log
p
(b
+
1)
+
1)
d
K
(b
+
1)
=
dim
Q
p
K[T
]/(T
b+1
)
.
Proof.
For
each
nonnegative
integer
j,
write
def
F
j
=
W
∩
{h
∈
K[T
]/(T
b+1
)
|
ord(h)
≥
p
j
−
1}
[so
F
j
is
a
Q
p
-vector
space,
and
F
j+1
⊆
F
j
].
Then
it
follows
immediately
from
our
assumption
that
W
\
{0}
⊆
F
that
for
each
nonnegative
integer
j,
dim
Q
p
(F
j
/F
j+1
)
≤
dim
Q
p
(K)
=
d
K
.
67
On
the
other
hand,
since
F
j
=
{0}
for
any
nonnegative
integer
j
such
that
p
j
>
p
j
−
1
≥
b
+
1,
we
thus
conclude
that
a
=
dim
Q
p
(W
)
=
+∞
dim
Q
p
(F
j
/F
j+1
)
≤
d
K
(log
p
(b
+
1)
+
1),
j=0
as
desired.
This
completes
the
proof
of
Lemma
2.15.
Theorem
2.16
(Existence
of
suitable
coverings).
In
the
notation
of
Propo-
sition
2.13,
suppose
further
that
K
is
a
p-adic
local
field.
Then
there
exists
a
def
def
real
number
C
g
Y
,d
K
that
depends
only
on
g
Y
=
g
Y
0
and
d
K
=
[K
:
Q
p
]
such
that
for
any
positive
integer
n
≥
C
g
Y
,d
K
,
after
possibly
replacing
K
by
a
finite
extension
field
of
K,
there
exist
•
a
[connected]
finite
étale
Galois
covering
W
n
→
Y
n
of
proper
hyperbolic
curves
over
K
of
degree
a
power
of
p,
•
a
semistable
model
W
n
of
W
n
over
O
K
,
•
a
morphism
ψ
:
W
n
→
Y
n
of
semistable
models
over
O
K
that
restricts
to
the
finite
étale
Galois
covering
W
n
→
Y
n
,
•
an
irreducible
component
D
of
(W
n
)
s
whose
normalization
is
of
genus
≥
1
such
that
ψ(D)
=
(y
n
)
s
∈
(Y
n
)
s
.
Proof.
Fix
a
positive
integer
n.
First,
we
consider
the
natural
homomorphisms
of
Z
p
-modules
∼
Hom
G
K
(T
qtr,n
,
Z
p
(1))
←
m
)
→
H
0
(
T
qtr,n
,
Ω
Hom
O
K
(
T
qtr,n
,
G
inv
T
qtr,n
)
0
(
T
cnn,n
,
Ω
T
cnn,n
)
→
H
inv
∼
←
H
0
(A
n
,
Ω
A
n
)
→
H
0
(U
y
n
,
Ω
Y
n
)
Y
,y
→
Ω
Y
n
,y
n
⊗
O
Y
n
,yn
O
n
n
2g
Yn
→
Ω
Y
n
,y
n,η
/m
y
n,η
Ω
Y
n
,y
n,η
∼
2g
Yn
→
K[t]/(t
)
dt
,
where
•
the
first
arrow
denotes
the
natural
bijective
homomorphism
of
Proposition
2.13,
(i);
68
0
•
“H
inv
(−)”
denotes
the
O
K
-submodule
of
“H
0
(−)”
that
consists
of
the
invariant
differentials
on
the
p-divisible
group
in
the
first
argument
of
“H
0
(−)”;
•
the
second
arrow
denotes
the
injection
obtained
by
pulling
back
the
in-
def
variant
differential
d
log(U
)
=
dU
U
on
G
m
;
•
the
third
arrow
denotes
the
injection
induced
by
χ
n
[cf.
Proposition
2.11;
[Tate],
§2.2,
Proposition
1;
[Tate],
§4.2,
Corollary
1];
•
the
fourth
arrow
denotes
the
natural
isomorphism;
•
the
fifth
arrow
denotes
the
homomorphism
of
O
K
-modules
obtained
by
pulling
back
the
differentials
via
the
composite
map
U
y
n
→
J
n
A
n
that
maps
y
n
to
the
origin
[cf.
Definition
2.7,
(ii),
(b);
Definition
2.7,
(iii);
Proposition
2.12];
•
the
sixth
arrow
denotes
the
natural
injection;
•
the
seventh
arrow
denotes
the
natural
restriction
morphism;
•
m
y
n,η
denotes
the
maximal
ideal
of
O
Y
n
,y
n,η
;
•
the
final
arrow
denotes
the
natural
isomorphism
determined
by
choosing
a
“local
coordinate”
t,
i.e.,
an
element
of
the
maximal
ideal
m
Y
n
,y
n
of
O
Y
n
,y
n
such
that
t
and
m
K
generate
m
Y
n
,y
n
.
Write
m
)
−→
O
K
[[t]]
×
Ψ
:
Hom
O
K
(
T
qtr,n
,
G
for
the
assignment
discussed
in
Proposition
2.13,
(ii)
[i.e.,
relative
to
the
local
coordinate
t
chosen
above];
∼
Yn
m
)
→
Ω
Y
,y
/m
2g
→
K[t]/(t
2g
Yn
)
dt
Ξ
:
Hom
O
K
(
T
qtr,n
,
G
y
n,η
Ω
Y
n
,y
n,η
n
n,η
for
the
injective
[by
Lemma
2.14]
composite
of
the
second
to
the
seventh
arrows
in
the
first
display
of
the
present
proof.
Thus,
Ξ
may
be
understood
as
the
result
of
composing
Ψ
with
the
operation
of
taking
the
logarithmic
derivative
with
respect
to
t
and
then
truncating
the
terms
of
degree
≥
2g
Y
n
.
Observe
that
so
far
we
have
not
applied
the
assumption
that
K
is
a
p-adic
local
field.
Now
we
proceed
to
apply
this
assumption.
Recall
from
Proposition
2.8,
(i),
(ii);
Proposition
2.11,
(ii)
[cf.
also
the
initial
portions
of
Proposition
2.6
and
Definition
2.7],
that
rank
Z
p
T
qtr,n
=
rank
Z
p
T
tor,n
=
p
n
(l
m
−
l
m−1
)e
X
;
2g
Y
n
=
2(p
n
(g
Y
0
−
1)
+
1),
where
e
X
≥
1.
Thus,
one
verifies
immediately
that
there
exists
a
real
number
C
g
Y
,d
K
that
depends
only
on
g
Y
=
g
Y
0
and
d
K
=
[K
:
Q
p
]
such
that
for
any
positive
integer
n
≥
C
g
Y
,d
K
,
it
holds
that
p
n
(l
m
−
l
m−1
)e
X
>
d
K
(log
p
(2(p
n
(g
Y
0
−
1)
+
1))
+
1).
69
In
particular,
we
conclude
from
Lemma
2.15
that
there
exists
a
homomorphism
m
)
such
that
Ξ(f
)
=
0,
and
ord(Ξ(f
))
+
1
is
not
a
[non-
f
∈
Hom
O
K
(
T
qtr,n
,
G
negative
integral]
power
of
p.
Fix
such
a
homomorphism
f
.
Then
note
that
it
follows
from
our
choice
of
f
that
we
may
write
a
i
t
i
∈
O
K
[[t]]
×
Ψ(f
)
=
1
+
i≥1
[cf.
Proposition
2.13,
(ii)],
where,
if
we
write
i
0
for
the
smallest
positive
integer
i
such
that
a
i
=
0,
then
i
0
is
not
a
[nonnegative
integral]
power
of
p.
In
the
following,
we
shall
apply
Proposition
1.6,
where
we
take
“g(t)”
to
be
Ψ(f
)
and
apply
the
isomorphism
of
topological
O
K
-algebras
∼
Y
,y
O
K
[[t]]
→
O
n
n
def
determined
by
t,
to
complete
the
proof
of
Theorem
2.16.
Write
N
=
μ
+
1,
where
μ
is
the
“μ”
that
results
from
applying
Proposition
1.6,
(i);
ψ
η
:
W
n
−→
Y
n
for
the
[connected]
finite
étale
Galois
covering
over
K
obtained
by
pulling-back
the
morphism
induced
by
multiplication
by
p
N
on
A
n
via
the
composite
mor-
phism
Y
n
→
U
y
n
→
J
n
A
n
[cf.
Proposition
2.12].
Next,
let
us
observe
that
it
follows
immediately
from
the
various
definitions
of
the
morphisms
involved
that
the
composite
morphism
φ
c
◦
λ
h
◦
τ
◦
φ
c
|
U
:
U
c
→
Spec
O
K
[[t]]
1
2
c
2
2
[cf.
the
two
commutative
diagrams
of
Proposition
1.6,
(i)],
together
with
the
∼
isomorphism
O
K
[[t]]
→
O
Y
n
,y
n
,
allow
one
to
regard
the
p-adic
completion
R
Y
of
Γ(O
U
c
2
,
U
c
2
)
at
the
generic
point
of
(U
c
2
)
s
as
the
p-adic
completion
of
the
ring
of
integers
R
Y
of
a
certain
discrete
residue-transcendental
p-valuation
on
the
function
field
of
Y
n
.
Write
R
W/Y
for
the
normalization
of
R
Y
in
the
function
field
of
W
n
.
Thus,
since
R
Y
is
[a
localization
of
a
ring
of
finite
type
over
the
complete
discrete
valuation
ring
O
K
,
hence]
excellent,
it
follows
that
R
W/Y
is
finite
over
R
Y
.
Next,
let
us
observe
that
it
follows
from
the
relations
ι
◦
λ
g
◦
φ
c
◦
λ
h
=
(p
μ
)
◦
ξ
g
,
1
ξ
g
◦
τ
◦
(φ
c
2
|
U
c
2
)
=
ι
◦
(φ
π
p
|
U
πp
)
◦
θ
g
,
ι
◦
(φ
π
p
|
U
πp
)
◦
θ
g
◦
f
Y
=
(p)
◦
ι
◦
(φ
π
|
U
π
)
◦
θ
Y
in
the
first
and
second
commutative
diagrams
of
Proposition
1.6,
(i),
and
the
commutative
diagram
of
Remark
1.6.1
that
we
obtain
relations
ι
◦
λ
g
◦
φ
c
◦
λ
h
◦
τ
◦
(φ
c
|
U
)
◦
f
Y
=
(p
μ
)
◦
ξ
g
◦
τ
◦
(φ
c
|
U
)
◦
f
Y
1
2
c
2
2
c
2
=
(p
)
◦
ι
◦
(φ
π
p
|
U
πp
)
◦
θ
g
◦
f
Y
μ
=
(p
μ
)
◦
(p)
◦
ι
◦
(φ
π
|
U
π
)
◦
θ
Y
,
where
70
•
the
composite
of
the
first
and
second
equalities
implies
that,
after
possi-
bly
replacing
K
by
a
finite
extension
field
of
K
[which
in
fact
may
be
Y
)
K
-
taken
to
be
unramified
—
cf.
Lemma
2.10,
(i)],
the
tautological
(
R
def
Y
)
K
=
R
Y
⊗
O
K,
lifts
to
an
valued
point
y
R
of
Y
n
,
where
we
write
(
R
K
∗
(
R
Y
)
K
-valued
point
y
R
of
a
certain
intermediate
covering
W
n
→
Y
n
∗
of
W
n
→
Y
n
that
corresponds
to
multiplication
by
p
μ
on
the
codomain
of
m
),
while
the
homomorphism
f
∈
Hom
O
K
(
T
qtr,n
,
G
•
the
third
equality
implies
[cf.
also
the
“essentially
cartesian”
nature
of
∗
the
squares
in
the
commutative
diagram
of
Remark
1.6.1]
that
y
R
lifts
def
W
)
K
=
R
W
⊗
O
K,
W
)
K
-valued
point
w
R
of
W
n
,
where
(
R
to
an
(
R
K
and
R
W
denotes
the
p-adic
completion
of
some
localization
R
W
of
R
W/Y
W
admits
a
at
a
maximal
ideal
of
R
W/Y
such
that
the
spectrum
of
R
tautological
isomorphism
over
R
Y
to
the
spectrum
of
the
p-adic
completion
of
“Γ(O
Y
,
Y
)”
at
the
generic
point
of
“Y
s
”
[i.e.,
where
the
quotation
marks
refer
to
the
notation
of
Proposition
1.6,
(ii)].
In
particular,
since
i
0
is
not
a
[nonnegative
integral]
power
of
p,
it
follows
from
W
,
hence
also
Proposition
1.6,
(ii),
and
Remark
1.6.2
that
the
residue
field
of
R
the
residue
field
of
R
W
,
is
the
function
field
of
a
curve
over
the
residue
field
of
O
K
of
genus
≥
1.
Thus,
we
conclude
from
Proposition
2.3,
(ii),
(iii),
that,
after
possibly
replacing
K
by
a
finite
extension
field
of
K,
there
exist
a
compactified
semistable
model
W
n
of
W
n
over
O
K
,
together
with
a
dominant
morphism
ψ
:
W
n
−→
Y
n
over
O
K
,
such
that
•
ψ
restricts
to
the
finite
étale
Galois
covering
ψ
η
:
W
n
→
Y
n
;
•
R
W
is
the
local
ring
of
W
n
at
the
generic
point
of
an
irreducible
component
D
of
(W
n
)
s
whose
normalization
is
of
genus
≥
1;
•
ψ(D)
=
(y
n
)
s
∈
(Y
n
)
s
.
This
completes
the
proof
of
Theorem
2.16.
Theorem
2.17
(Resolution
of
nonsingularities
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields).
Let
Σ
⊆
Primes
be
a
subset
of
cardinality
≥
2;
K
a
p-adic
local
field,
for
some
p
∈
Σ;
X
a
hyperbolic
curve
over
K;
L
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p
that
contains
K
as
a
topological
subfield.
Then
X
L
satisfies
Σ-RNS
if
and
only
if
the
residue
field
of
L
is
algebraic
over
the
finite
field
of
cardinality
p.
Proof.
First,
we
observe
that
it
follows
formally
from
Remark
2.2.3,
(v),
that
it
suffices
to
verify
that
X
satisfies
Σ-RNS.
Next,
we
observe
that
it
follows
71
immediately
from
the
various
definitions
involved
that
we
may
assume
without
loss
of
generality
that
X
has
stable
reduction
over
K.
Write
X
for
the
[unique,
up
to
unique
isomorphism]
compactified
stable
model
of
X
over
O
K
.
Then,
in
light
of
Propositions
2.3,
(xii);
2.4,
(i),
(ii),
by
replacing
X
by
the
[unique,
up
to
unique
isomorphism]
smooth
compactification
of
a
suitable
connected
geometrically
pro-Σ
finite
étale
covering
of
X,
we
may
assume
without
loss
of
generality
that:
•
X
is
a
proper
hyperbolic
curve
over
K,
•
X
s
is
split,
•
X
s
is
singular,
and
•
every
irreducible
component
of
X
s
is
a
smooth
curve
of
genus
≥
2.
In
particular,
X
now
satisfies
the
assumptions
imposed
in
the
respective
initial
portions
of
Proposition
2.6
and
Definition
2.7.
Next,
observe
that
since
the
covering
Y
n
→
Y
is
combinatorial
[cf.
Definition
2.7,
(ii)],
it
follows
immediately
that
this
covering
induces
a
surjection
Y
n
(K)
Y
(K)
on
K-rational
points.
Thus,
it
follows
immediately
from
Proposition
2.4,
(iii),
and
Theorem
2.16
that
X
satisfies
Σ-RNS.
This
completes
the
proof
of
Theorem
2.17.
3
Point-theoreticity,
metric-admissibility,
and
arith-
metic
cuspidalization
Let
p
be
a
prime
number.
In
the
present
section,
we
first
recall
the
well-
known
classification
of
the
points
of
the
topological
Berkovich
space
associated
to
a
proper
hyperbolic
curve
over
a
mixed
characteristic
complete
discrete
val-
uation
field
via
the
notion
of
type
i
points,
where
i
∈
{1,
2,
3,
4}
[cf.
Defi-
nition
3.1].
Next,
we
introduce
a
certain
combinatorial
classification
of
the
VE-chains
considered
in
§2
[cf.
Definition
3.2]
and
observe
that
this
classifica-
tion
of
VE-chains
leads
naturally
to
a
purely
combinatorial
characterization
of
the
well-known
classification
via
type
i
points
mentioned
above
[cf.
Proposi-
tions
3.3,
3.4].
This
combinatorial
classification/characterization
[cf.
also
the
approach
of
Propositions
3.7,
3.8]
was
motivated
by
the
argument
applied
in
the
proof
of
[CbTpIV],
Theorem
A.7.
We
then
apply
the
theory
of
§2
to
give
a
group-theoretic
characterization,
motivated
by
[but
by
no
means
identical
to]
the
characterization
of
[Lpg2],
§4,
of
the
type
i
points
in
terms
of
the
geometric
Σ-tempered
fundamental
group
of
the
hyperbolic
curve
[cf.
Propositions
3.5,
3.9].
Then,
by
combining
this
group-theoretic
characterization
with
[AbsTopII],
Corollary
2.9,
we
prove
an
absolute
version
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
over
p-adic
local
fields
[cf.
Theorem
3.12].
This
settles
one
of
the
major
open
questions
in
anabelian
geometry.
As
a
corollary
of
this
absolute
version
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
over
p-adic
local
72
fields,
together
with
[HMM],
Theorem
A,
we
also
obtain
an
absolute
version
of
the
Grothendieck
Conjecture
for
configuration
spaces
associated
to
hyper-
bolic
curves
over
p-adic
local
fields
[cf.
Theorem
3.13].
We
then
switch
gears
to
discuss
metric-admissibility
for
p-adic
hyperbolic
curves.
This
discussion
of
metric-admissibility
leads
to
a
proof
that
all
of
the
various
p-adic
versions
of
the
Grothendieck-Teichmüller
group
that
appear
in
the
literature
in
fact
coin-
cide
[cf.
Theorem
3.16].
Moreover,
as
an
application
of
Corollary
2.5,
(i),
and
the
theory
developed
in
the
present
section,
together
with
the
theory
of
metric-
admissibility
developed
in
[CbTpIII],
§3,
we
obtain
a
construction
of
a
certain
type
of
arithmetic
cuspidalization
of
the
[Primes-]
tempered
fundamental
group
of
a
hyperbolic
curve
over
Q
p
[cf.
Theorem
3.20].
Definition
3.1.
Let
Σ
⊆
Primes
be
a
nonempty
subset;
K
a
mixed
charac-
teristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
proper
→
X
a
universal
geometrically
pro-Σ
covering
of
X.
hyperbolic
curve
over
K;
X
Write
Ω
for
the
p-adic
completion
of
[some
fixed]
K;
(−)
an
for
the
topological
an
[cf.
Berkovich
space
associated
to
(−).
Let
x
∈
X
an
be
an
element;
x̃
∈
X
Proposition
2.3,
(vii),
(viii)]
a
lifting
of
x.
Then:
(i)
We
shall
say
that
x̃
is
of
type
1
if
x̃
is
determined
by
a
point-theoretic
of
X
associated
to
some
point
p-valuation
on
the
function
field
K(
X)
∈
X(Ω)
[cf.
Definition
2.2,
(ii)].
(ii)
We
shall
say
that
x̃
is
of
type
2
if
x̃
is
determined
by
an
inverse
system
of
discrete
residue-transcendental
p-valuations
associated
to
irreducible
components
of
the
special
fibers
of
[compactified]
semistable
models
with
split
reduction
of
the
domain
curves
of
connected
finite
étale
coverings
→
Z
→
X
[cf.
Proposition
2.3,
Z
→
X
equipped
with
a
factorization
X
(ii),
(iii)].
(iii)
We
shall
say
that
x̃
is
of
type
3
if
there
exist
a
finite
extension
field
L
of
K,
a
[compactified]
semistable
model
with
split
reduction
X
of
X
L
over
O
L
,
and
a
node
e
of
X
s
such
that
x̃
arises
as
the
inverse
image
of
a
lifting
tor
of
some
element
∈
D
e
⊆
VE(X
)
tor
[cf.
Definition
2.2,
(vi)]
∈
VE(
X)
∼
an
→
tor
[cf.
Proposition
2.3,
(viii)].
via
the
homeomorphism
X
VE(
X)
(iv)
We
shall
say
that
x̃
is
of
type
4
if,
for
each
i
∈
{1,
2,
3},
x̃
is
not
of
type
i.
(v)
For
each
i
∈
{1,
2,
3,
4},
we
shall
say
that
x
is
of
type
i
if
x̃
is
of
type
i.
[One
verifies
immediately
that,
for
each
i
∈
{1,
2,
3,
4},
the
condition
that
x
is
of
type
i
is
independent
of
the
choices
of
Σ
and
x̃.]
For
each
an
[i]
⊆
X
an
)
i
∈
{1,
2,
3,
4},
we
shall
write
X
an
[i]
⊆
X
an
(respectively,
X
an
).
for
the
subset
of
points
of
type
i
of
X
an
(respectively,
X
73
Remark
3.1.1.
In
the
notation
of
Definition
3.1,
we
observe
that
an
=
X
an
[1]
∪
X
an
[2]
∪
X
an
[3]
∪
X
an
[4];
X
X
an
=
X
an
[1]
∪
X
an
[2]
∪
X
an
[3]
∪
X
an
[4],
and,
moreover,
for
each
pair
of
distinct
i,
j
∈
{1,
2,
3,
4},
an
[j]
=
∅;
an
[i]
∩
X
X
X
an
[i]
∩
X
an
[j]
=
∅.
an
[j]
=
∅.
First,
by
considering
an
[i]
∩
X
Indeed,
it
suffices
to
verify
that
X
the
residue
fields
of
the
valuation
rings
under
consideration,
we
conclude
that
an
[2]
=
∅.
Next,
we
observe
that
it
follows
immediately
from
the
an
[1]
∩
X
X
discussion
of
the
construction
of
“VE(Z)
tor
”
in
Definition
2.2,
(vi),
that,
for
an
[i]
∩
X
an
[3]
=
∅.
Finally,
we
observe
that
it
is
a
tautology
each
i
∈
{1,
2},
X
an
[i]
∩
X
an
[4]
=
∅.
This
completes
the
proof
of
that,
for
each
i
∈
{1,
2,
3},
X
relations
in
the
above
display.
Definition
3.2.
We
maintain
the
notation
of
Definition
3.1.
Let
c
=
(c
Z
)
Z∈S
∈
where
S
denotes
the
directed
set
of
[compactified]
semistable
models
VE(
X),
[cf.
Definition
2.2,
(iii)].
Write
V
c
⊆
S
that
appear
in
the
definition
of
VE(
X)
(respectively,
E
c
⊆
S)
for
the
subset
of
[compactified]
semistable
models
Z
such
that
c
Z
is
a
vertex
(respectively,
an
edge).
Then:
(i)
We
shall
say
that
c
is
asymptotically
verticial
(respectively,
asymptotically
edge-like)
if
the
subset
V
c
⊆
S
(respectively,
E
c
⊆
S)
forms
a
cofinal
subset
of
S.
[In
particular,
if
c
is
asymptotically
verticial
(respectively,
asymptotically
edge-like),
then
V
c
(respectively,
E
c
)
may
be
regarded
as
a
directed
set
in
a
natural
way.]
(ii)
Suppose
that
c
is
asymptotically
verticial.
Then
we
shall
say
that
c
is
strongly
verticial
if
there
exists
a
cofinal
subset
S
c
⊆
V
c
satisfying
the
following
condition:
Let
Z
1
,
Z
2
∈
S
c
be
distinct
elements
such
that
Z
2
dominates
Z
1
.
Then
the
generic
point
of
the
irreducible
component
of
(Z
2
)
s
that
corresponds
to
c
Z
2
maps
to
the
generic
point
of
the
irreducible
component
of
(Z
1
)
s
that
corresponds
to
c
Z
1
via
the
dominant
morphism
Z
2
→
Z
1
.
(iii)
Suppose
that
c
is
asymptotically
verticial.
Then
we
shall
say
that
c
is
weakly
verticial
if
there
exists
a
cofinal
subset
S
c
⊆
V
c
satisfying
the
following
condition:
Let
Z
1
,
Z
2
∈
S
c
be
distinct
elements
such
that
Z
2
dominates
Z
1
.
Then
the
generic
point
of
the
irreducible
component
of
(Z
2
)
s
that
corresponds
to
c
Z
2
maps
to
a
closed
point
in
the
interior
of
the
irreducible
component
of
(Z
1
)
s
that
corresponds
to
c
Z
1
via
the
dominant
morphism
Z
2
→
Z
1
.
74
(iv)
Suppose
that
c
is
asymptotically
edge-like.
Then
we
shall
say
that
c
is
weakly
edge-like
if
there
exists
a
cofinal
subset
S
c
⊆
E
c
satisfying
the
following
condition:
Let
Z
1
,
Z
2
∈
S
c
be
distinct
elements
such
that
Z
2
dominates
Z
1
.
Then
there
exists
a
toral
compactified
semistable
model
Z
1
∗
relative
to
Z
1
such
that
the
dominant
morphism
Z
2
→
Z
1
admits
a
factorization
Z
2
→
Z
1
∗
→
Z
1
,
and
the
node
of
(Z
2
)
s
that
corresponds
to
c
Z
2
maps
to
a
closed
point
in
the
interior
of
an
irreducible
component
of
(Z
1
∗
)
s
[that
necessarily
lies
over
the
node
of
(Z
1
)
s
that
corresponds
to
c
Z
1
]
via
the
dominant
morphism
Z
2
→
Z
1
∗
.
(v)
Suppose
that
c
is
asymptotically
edge-like.
Then
we
shall
say
that
c
is
strongly
edge-like
if
there
exists
a
cofinal
subset
S
c
⊆
E
c
satisfying
the
following
condition:
Let
Z
1
,
Z
2
∈
S
c
be
distinct
elements
such
that
Z
2
dominates
Z
1
.
Then,
for
each
toral
compactified
semistable
model
Z
1
∗
relative
to
Z
1
that
admits
a
factorization
Z
2
→
Z
1
∗
→
Z
1
,
the
node
of
(Z
2
)
s
that
corresponds
to
c
Z
2
maps
to
a
node
of
(Z
1
∗
)
s
[that
necessarily
lies
over
the
node
of
(Z
1
)
s
that
corresponds
to
c
Z
1
]
via
the
dominant
morphism
Z
2
→
Z
1
∗
.
(vi)
We
shall
write
vtc
⊆
VE(
X);
VE(
X)
edg
⊆
VE(
X);
VE(
X)
str-vtc
⊆
VE(
X);
VE(
X)
wk-vtc
⊆
VE(
X);
VE(
X)
str-edg
⊆
VE(
X);
VE(
X)
wk-edg
⊆
VE(
X),
VE(
X)
respectively,
for
the
subsets
of
asymptotically
verticial
VE-chains,
asymp-
totically
edge-like
VE-chains,
strongly
verticial
VE-chains,
weakly
verti-
cial
VE-chains,
strongly
edge-like
VE-chains,
and
weakly
edge-like
VE-
chains.
Also,
for
each
∈
{vtc,
edg,
str-vtc,
wk-vtc,
str-edg,
wk-edg},
we
shall
write
prim,
=
VE(
X)
prim
∩
VE(
X)
VE(
X)
def
(⊆
VE(
X)).
(vii)
Let
Z
∈
S.
Then
in
the
notation
of
Definition
2.2,
(vi),
we
shall
write
def
VE(Z)
tor,rat
=
V
w
(⊆
VE(Z)
tor
);
D
e
(⊆
VE(Z)
tor
).
w∈V(Z)
def
VE(Z)
tor,irr
=
e∈E(Z)
75
Note
that
one
verifies
immediately
that
VE(Z)
tor
=
VE(Z)
tor,rat
VE(Z)
tor,irr
,
and
that,
for
any
Z
1
,
Z
2
∈
S
such
that
Z
2
dominates
Z
1
,
the
natural
map
VE(Z
2
)
tor
→
VE(Z
1
)
tor
induces
a
map
VE(Z
2
)
tor,rat
→
VE(Z
1
)
tor,rat
[cf.
the
discussion
of
Definition
2.2,
(vi)].
Remark
3.2.1.
In
the
notation
of
Definition
3.2,
we
observe
that
it
follows
im-
mediately
from
the
various
definitions
involved
[cf.
also
Remarks
2.1.4,
2.1.5;
Definition
2.2,
(vi)]
that:
str-vtc
∩
VE(
X)
wk-vtc
=
∅;
VE(
X)
str-edg
∩
VE(
X)
wk-edg
=
∅;
VE(
X)
vtc
=
VE(
X)
str-vtc
∪
VE(
X)
wk-vtc
;
VE(
X)
edg
=
VE(
X)
str-edg
∪
VE(
X)
wk-edg
;
VE(
X)
=
VE(
X)
vtc
∪
VE(
X)
edg
.
VE(
X)
Proposition
3.3
(Elementary
properties
of
the
combinatorial
classifi-
cation
of
VE-chains).
In
the
notation
of
Definition
3.2,
by
forming
the
in-
ductive
limit
of
the
natural
log
structures
on
the
[compactified]
semistable
models
Z
∈
S,
we
obtain
an
ind-log
structure
on
the
pro-scheme
lim
Z∈S
Z.
Then
the
←−
following
hold:
(i)
Let
c
∈
VE(
X).
Write
z̃
c
for
the
center
on
lim
Z∈S
Z
of
the
valuation
←−
ring
R
c
associated
to
c
[cf.
Proposition
2.3,
(vi)];
M
c
pf
for
the
perfection
of
the
inductive
limit
monoid
obtained
by
forming
the
stalk
at
z̃
c
of
the
characteristic
of
the
ind-log
structure
on
the
pro-scheme
lim
Z∈S
Z.
Then,
←−
str-edg
),
then
M
pf
is
isomorphic
vtc
(respectively,
c
∈
VE(
X)
if
c
∈
VE(
X)
c
to
Q
≥0
(respectively,
Q
≥0
×
Q
≥0
).
(ii)
The
following
relations
hold:
str-vtc
∩
VE(
X)
str-edg
=
∅;
VE(
X)
str-vtc
∩
VE(
X)
wk-vtc
=
∅;
VE(
X)
wk-vtc
∩
VE(
X)
str-edg
=
∅;
VE(
X)
wk-edg
⊆
VE(
X)
wk-vtc
;
VE(
X)
=
VE(
X)
str-vtc
∪
VE(
X)
wk-vtc
∪
VE(
X)
str-edg
.
VE(
X)
76
Proof.
Assertion
(i)
follows
immediately
from
the
well-known
log
structure
of
M
c
pf
[cf.
also
the
discussion
of
the
subsection
in
Notations
and
Conventions
en-
titled
“Log
schemes”],
together
with
the
various
definitions
involved.
Assertion
(ii)
follows
immediately
from
assertion
(i),
together
with
the
various
defini-
tions
involved
[cf.
also
Remark
3.2.1].
This
completes
the
proof
of
Proposition
3.3.
Proposition
3.4
(Characterization
of
points
of
type
2
and
3
via
the
combinatorial
classification
of
VE-chains).
We
maintain
the
notation
of
Definition
3.2.
Then
the
following
hold:
\
VE(
X)
prim
.
Then
nonprim
def
=
VE(
X)
(i)
Write
VE(
X)
nonprim
∪
VE(
X)
prim,str-edg
.
str-edg
=
VE(
X)
VE(
X)
Moreover,
the
unique
nontrivial
generization
of
a
nonprimitive
VE-chain
[cf.
Proposition
2.3,
(x)]
is
strongly
verticial.
(ii)
For
Z
∈
S,
write
τ
X
str-edg
str-edg
⊆
VE(
X)
−→
tor
−→
VE(Z)
tor
:
VE(
X)
VE(
X)
τ
X,Z
for
the
natural
composite
map
[cf.
Proposition
2.3,
(viii)].
Then
for
any
str-edg
induces
a
map
Z
∈
S,
τ
X,Z
nonprim
→
VE(Z)
tor,rat
VE(
X)
prim,str-edg
,
there
exists
an
element
[cf.
(i)].
Moreover,
for
each
c
∈
VE(
X)
Z
c
∈
S
such
that
for
every
Z
∈
S
that
dominates
Z
c
,
str-edg
(c)
∈
VE(Z)
tor,irr
τ
X,Z
[cf.
(i)].
Finally,
Z
c
∈
S
may
be
taken
to
be
a
[compactified]
semistable
model
with
split
reduction
of
X
L
over
O
L
for
some
finite
extension
field
L
of
K.
∼
an
prim
→
X
[cf.
Proposition
2.3,
(viii)]
determines
(iii)
The
bijection
VE(
X)
bijections
∼
an
[2];
prim,str-vtc
→
X
VE(
X)
∼
prim,str-edg
→
X
an
[3].
VE(
X)
(iv)
Let
Y
be
a
proper
hyperbolic
curve
over
K;
f
:
Y
→
X
a
dominant
def
morphism
over
K;
y
∈
Y
an
.
Write
x
=
f
(y)
∈
X
an
.
Then,
for
each
i
∈
{1,
2,
3,
4},
y
is
of
type
i
if
and
only
if
x
is
of
type
i.
77
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
str-edg
.
First,
suppose
that
Next,
we
verify
assertion
(ii).
Let
c
∈
VE(
X)
nonprim
prim,str-vtc
c
∈
VE(
X)
.
Write
c
∈
VE(
X)
for
the
unique
nontrivial
gener-
ization
of
c
[cf.
assertion
(i)].
Then
it
follows
immediately
from
the
definition
of
τ
X
in
the
proof
of
Proposition
2.3,
(viii),
that
τ
X
(c)
=
τ
X
(c
).
In
particular,
it
prim,str-vtc
that
τ
(c
)
maps
to
follows
immediately
from
the
fact
that
c
∈
VE(
X)
X
str-edg
(c)
∈
VE(Z)
tor,rat
,
an
element
of
VE(Z)
tor,rat
for
any
Z
∈
S,
hence
that
τ
X,Z
prim,str-edg
.
Let
Z
c
∈
S
c
for
some
as
desired.
Next,
suppose
that
c
∈
VE(
X)
“S
c
”
as
in
Definition
3.2,
(v).
Let
Z
∈
S
be
an
element
that
dominates
Z
c
.
Then
one
verifies
immediately
[cf.
also
the
final
portion
of
Definition
3.2,
(vii)]
str-edg
(c)
∈
VE(Z)
tor,rat
implies
a
contradiction
to
the
condi-
that
any
relation
τ
X
str-edg
tion
of
Definition
3.2,
(v).
Thus,
we
conclude
that
τ
X,Z
(c)
∈
VE(Z)
tor,irr
,
as
desired.
The
fact
that
Z
c
∈
S
may
be
taken
to
be
a
[compactified]
semistable
model
with
split
reduction
of
X
L
over
O
L
for
some
finite
extension
field
L
of
K
follows
immediately
from
Proposition
2.3,
(iii),
(iv).
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
assertion
(ii),
together
with
the
various
definitions
involved
[cf.
also
the
final
portion
of
Definition
3.2,
(vii);
Proposition
3.3,
(ii)].
Finally,
we
consider
assertion
(iv).
First,
we
observe
that
the
asserted
equivalence
follows
immediately
from
the
various
definitions
involved
[cf.
also
Proposition
2.3,
(ii),
(iii),
in
the
case
where
i
=
2]
when
i
∈
{1,
2}.
Thus,
it
suffices
to
verify
the
asserted
equivalence
when
i
=
3.
When
i
=
3,
sufficiency
follows
immediately,
in
light
of
assertion
(iii)
and
Proposition
3.3,
(ii),
from
Definition
3.1,
(iii)
[cf.
also
the
discussion
of
Definition
2.2,
(vi)].
On
the
other
hand,
it
follows
immediately,
by
replacing
Y
by
the
normalization
of
Y
in
the
Galois
closure
of
the
finite
extension
of
function
fields
determined
by
f
and
applying
the
sufficiency
that
has
already
been
verified,
that
to
verify
necessity
when
i
=
3,
we
may
assume
without
loss
of
generality
that
the
finite
extension
of
function
fields
determined
by
f
is
Galois.
But
then
the
desired
necessity
follows
immediately
from
the
final
portion
of
assertion
(ii),
together
with
Proposition
2.3,
(iii),
(iv)
[cf.
also
the
discussion
of
Definition
2.2,
(vi)].
This
completes
the
proof
of
Proposition
3.4.
Proposition
3.5
(Types
of
points
and
geometrically
pro-l
decomposi-
tion
groups).
In
the
notation
of
Definition
3.2,
let
l
∈
Σ
\
{p}.
Suppose
that
X
satisfies
Σ-RNS.
Then
the
following
hold:
(i)
Let
wk-vtc
;
c
∈
VE(
X)
str-edg
).
str-vtc
(respectively,
c
∈
VE(
X)
c
∈
VE(
X)
→)
Z
→
For
each
connected
geometrically
pro-Σ
finite
étale
covering
(
X
def
{l}
X,
write
D
Z,c
⊆
Δ
lZ
=
Δ
Z
for
the
decomposition
subgroup
associated
to
c
of
the
geometric
pro-l
fundamental
group
of
Z
[cf.
the
subsection
in
78
Notations
and
Conventions
entitled
“Fundamental
groups”].
Then
there
→)
Y
→
X
exists
a
connected
geometrically
pro-Σ
finite
étale
covering
(
X
of
X
such
that,
for
each
connected
geometrically
pro-Σ
finite
étale
covering
→)
Z
→
Y
of
Y
,
D
Z,c
is
isomorphic
to
a
[nonabelian]
pro-l
surface
(
X
group
[cf.
[MT],
Definition
1.2]
(respectively,
the
trivial
group;
Z
l
).
an
and
the
subset
X
an
[i]
⊆
X
an
may
(ii)
Let
i
∈
{2,
3}.
Then
the
set
X
be
reconstructed,
functorially
with
respect
to
isomorphisms
of
topological
groups,
from
the
underlying
topological
group
of
the
geometric
Σ-tempered
fundamental
group
of
X
[cf.
the
subsection
in
Notations
and
Conventions
entitled
“Fundamental
groups”].
(iii)
Let
Y
,
Z
be
[not
necessarily
proper!]
hyperbolic
curves
over
K
that
satisfy
Σ-RNS;
Y
→
Y
,
Z
→
Z
universal
geometrically
pro-Σ
coverings
of
Y
,
Z,
respectively;
f
:
Y
→
Z
a
dominant
morphism
over
K;
H
⊆
G
K
a
closed
subgroup
such
that
the
restriction
to
H
of
the
l-adic
cyclotomic
charac-
ter
of
K
has
open
image,
and,
moreover,
the
intersection
H
∩I
K
of
H
with
the
inertia
subgroup
I
K
of
G
K
admits
a
surjection
to
[the
profinite
group]
(Σ)
def
(Σ)
def
=
Gal(
Y
/Y
),
s
Z
:
H
→
Π
=
Gal(
X/X)
sections
Z
l
;
s
Y
:
H
→
Π
Y
Z
(Σ)
of
the
restrictions
to
H
of
the
respective
natural
surjections
Π
Y
G
K
,
(Σ)
(Σ)
Π
Z
G
K
such
that
s
Y
is
mapped,
up
to
Π
Z
-conjugation,
by
f
to
s
Z
via
the
map
induced
by
f
on
geometrically
pro-Σ
fundamental
groups.
Write
Ω
H
⊆
Ω
for
the
subfield
of
Ω
fixed
by
H.
Then
s
Y
arises
from
a(n)
[necessarily
unique]
Ω
H
-rational
point
∈
Y
(Ω
H
)
if
and
only
if
s
Z
arises
from
a(n)
[necessarily
unique]
Ω
H
-rational
point
∈
Z(Ω
H
).
Proof.
Since
X
satisfies
Σ-RNS
[cf.
Definition
2.2,
(vii)],
assertion
(i)
follows
immediately
from
the
well-known
structure
of
the
maximal
pro-l
quotient
of
the
admissible
fundamental
group
of
a
stable
curve
over
a
separably
closed
field
of
characteristic
p
[cf.
e.g.,
[SemiAn],
Example
2.10],
together
with
the
various
def-
initions
involved.
Assertion
(ii)
follows
immediately
from
assertion
(i),
together
with
Corollary
2.5,
(i);
Proposition
3.4,
(iii)
[cf.
also
Proposition
3.3,
(ii)].
Fi-
nally,
we
consider
assertion
(iii).
First,
we
observe
that
it
follows
immediately
(Σ)
from
the
profinite
nature
of
the
topological
group
Π
Y
that,
by
replacing
Y
and
Z
by
the
smooth
compactifications
of
the
various
finite
étale
coverings
of
Y
and
Z
corresponding,
respectively,
to
suitable
open
neighborhoods
of
the
image
of
(Σ)
(Σ)
s
Y
in
Π
Y
×
G
K
H
and
the
image
of
s
Z
in
Π
Z
×
G
K
H,
we
may
assume
without
loss
of
generality
that
Y
and
Z
are
proper.
Next,
we
observe
that
the
various
uniqueness
assertions
in
the
statement
of
Proposition
3.5,
(iii),
follow
immedi-
ately
from
the
final
portion
of
Proposition
2.4,
(vii),
and
that
necessity
follows
immediately
from
the
various
definitions
involved.
Thus,
it
suffices
to
verify
sufficiency.
On
the
other
hand,
sufficiency
follows,
in
light
of
the
equivalences
of
Proposition
3.4,
(iv),
formally
from
the
final
portion
of
Proposition
2.4,
(vii).
This
completes
the
proof
of
Proposition
3.5.
79
Definition
3.6.
Let
K
be
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
proper
hyperbolic
curve
over
K.
Write
K(X
K
)
for
the
function
field
of
X
K
.
Let
v
be
a
p-valuation
on
K(X
K
).
Write
(O
K
⊆)
O
v
⊆
K(X
K
)
for
the
valuation
ring
associated
to
v.
[Note
that
it
follows
immediately
from
the
well-known
theory
of
one-dimensional
function
def
fields
that
(O
v
)
K
=
O
v
·
K
⊆
K(X
K
)
is
equal
either
to
K(X
K
)
or
to
the
discrete
valuation
ring
associated
to
a
closed
point
of
X
K
.]
Then:
(i)
Let
M
be
an
O
v
-module.
Then
we
shall
say
that
M
is
bounded
if
the
image
def
of
M
via
the
natural
morphism
M
→
M
K
=
M
⊗
O
K
K
is
contained
in
a
finitely
generated
O
v
-submodule
of
M
K
.
We
shall
say
that
M
is
unbounded
if
M
is
not
bounded.
(ii)
We
shall
say
that
the
p-valuation
v
is
differentially
bounded
(respectively,
differentially
unbounded)
if
the
O
v
-module
of
relative
differentials
Ω
O
v
/O
K
is
bounded
(respectively,
unbounded).
Proposition
3.7
(Approximation
of
closed
points
of
the
generic
fiber
via
generic
points
of
special
fibers).
Let
K
be
a
mixed
characteristic
com-
plete
discrete
valuation
field
of
residue
characteristic
p;
K
=
K
0
⊆
K
1
⊆
·
·
·
⊆
K
i
⊆
·
·
·
an
ascending
chain
of
finite
extension
fields
of
K
contained
in
K
and
indexed
by
N.
Write
Ω
for
the
p-adic
completion
of
K.
Let
X
be
a
hyperbolic
curve
over
K.
For
each
i
∈
N,
let
X
i
be
a
compactified
semistable
model
with
split
reduction
of
X
K
i
over
O
K
i
;
φ
i+1
:
X
i+1
−→
X
i
×
O
Ki
O
K
i+1
a
dominant
morphism
over
O
K
i+1
that
induces
the
identity
automorphism
on
the
generic
fiber;
v
i
an
irreducible
component
of
(X
i
)
s
.
Suppose
that,
for
each
i
∈
N,
the
projection
to
X
i
of
φ
i+1
(v
i+1
)
is
a
closed
point
x
i
∈
(X
i
)
s
⊆
X
i
of
(X
i
)
s
that
lies
in
the
smooth
locus
of
v
i
.
Then
the
following
hold:
(i)
For
each
i
∈
N,
let
ψ
i
:
Spec
O
K
i
→
X
i
be
a
section
whose
image
contains
def
x
i
.
Then
there
exists
a
collection
{t
i
,
γ
i+1
,
π
i+1
}
i∈N
of
elements
t
i
∈
A
i
=
O
X
i
,x
i
and
γ
i+1
,
π
i+1
∈
m
K
i+1
such
that,
for
each
i
∈
N,
t
i
∈
A
i
is
a
generator
of
the
ideal
that
defines
the
scheme-theoretic
image
of
ψ
i
,
and
t
i
=
γ
i+1
+
π
i+1
t
i+1
,
where
we
regard
A
i
as
a
subring
of
A
i+1
via
the
injection
A
i
→
A
i+1
induced
by
the
composite
X
i+1
→
X
i
[which
maps
x
i+1
→
x
i
]
of
φ
i+1
with
the
projection
to
X
i
.
80
def
(ii)
We
maintain
the
notation
of
(i).
For
each
positive
integer
i,
write
l
i
=
v
p
(π
i
).
Suppose
that
the
equality
l
i
=
+∞
i≥1
holds.
Then
there
exists
a
closed
point
x
Ω
of
X
Ω
such
that,
for
each
i
∈
N,
the
center
on
X
i
of
the
closed
point
x
Ω
of
X
Ω
,
hence
also
of
the
point-theoretic
p-valuation
on
the
function
field
of
X
K
i
determined
by
the
closed
point
x
Ω
of
X
Ω
,
coincides
with
x
i
.
Proof.
First,
we
verify
assertion
(i).
We
construct
elements
t
i
∈
A
i
and
γ
i+1
,
π
i+1
∈
m
K
i+1
,
for
i
∈
N,
by
induction
on
i
∈
N.
Let
t
0
∈
A
0
be
a
generator
of
the
ideal
that
defines
the
scheme-theoretic
image
of
ψ
0
;
i
∈
N.
Suppose
that
the
elements
t
j+1
,
γ
j+1
,
and
π
j+1
have
been
constructed
for
j
∈
N
such
that
j
<
i.
Write
γ
i+1
∈
m
K
i+1
for
the
image
of
t
i
via
the
composite
homomor-
phism
A
i
⊆
A
i+1
→
O
K
i+1
induced
by
ψ
i+1
.
Next,
observe
that
the
image
in
A
i+1
⊗
O
Ki+1
K
i+1
of
t
i
−
γ
i+1
is
a
generator
of
the
maximal
ideal
associated
to
the
closed
point
of
X
K
i+1
determined
by
ψ
i+1
.
Thus,
since
x
i+1
lies
in
the
smooth
locus
of
(X
i+1
)
s
,
and
A
i+1
is
a
regular
local
ring,
hence
a
unique
fac-
torization
domain,
we
conclude
that
there
exists
an
element
π
i+1
∈
m
K
i+1
such
that
t
i
−γ
i+1
=
π
i+1
t
i+1
for
some
generator
t
i+1
∈
A
i+1
of
the
ideal
that
defines
the
scheme-theoretic
image
of
ψ
i+1
.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Suppose
that
i≥1
l
i
=
+∞.
For
each
i
∈
N,
fix
elements
t
i
∈
A
i
and
γ
i+1
,
π
i+1
∈
m
K
i+1
as
in
the
statement
of
assertion
(i).
For
each
positive
integer
j,
write
def
s
j
=
γ
j
·
π
i
.
1≤i≤j−1
it
follows
immediately
from
our
assump-
Then
since
O
Ω
is
p-adically
complete,
tion
that
i≥1
l
i
=
+∞
that
j≥1
s
j
converges
to
an
element
γ
∈
m
Ω
.
Write
x
Ω
for
the
closed
point
of
X
Ω
determined
by
the
homomorphism
ψ
Ω
:
A
0
→
O
Ω
over
O
K
0
that
maps
t
0
→
γ.
Now
observe
that
it
follows
immediately
from
the
definition
of
γ
that,
for
each
i
∈
N,
ψ
Ω
extends
uniquely
to
a
homomorphism
A
i
→
O
Ω
over
O
K
i
.
On
the
other
hand,
the
existence
of
such
unique
exten-
sions
implies
that
the
center
on
X
i
of
the
closed
point
x
Ω
of
X
Ω
,
hence
also
of
the
point-theoretic
p-valuation
on
the
function
field
of
X
K
i
determined
by
the
closed
point
x
Ω
of
X
Ω
,
coincides
with
x
i
.
This
completes
the
proof
of
assertion
(ii),
hence
of
Proposition
3.7.
Proposition
3.8
(Characterization
of
points
of
type
1
via
differentially
unboundedness).
In
the
notation
of
Definition
3.6
[cf.
also
the
notation
of
(⊇
K(X
))
that
restricts
Proposition
2.3,
(viii)],
let
v
be
a
p-valuation
of
K(
X)
K
to
v
on
K(X
K
)
[cf.
Remark
2.2.4].
Suppose
that
v
is
primitive.
Then
the
point
an
associated
to
v
[cf.
Proposition
2.3,
(viii)]
is
of
type
1
if
and
only
if
x
v
∈
X
v
is
differentially
unbounded.
81
an
.
Note
that,
in
Proof.
Write
x
v
∈
X
an
for
the
point
determined
by
x
v
∈
X
light
of
Remark
3.1.1,
it
suffices
to
verify
that
if
x
v
is
of
type
1
(respectively,
of
type
i
∈
{2,
3,
4}),
then
v
is
differentially
unbounded
(respectively,
differentially
bounded).
First,
we
verify
that
if
x
v
is
of
type
1,
then
v
is
differentially
unbounded.
Suppose
that
x
v
is
of
type
1.
Write
Ω
for
the
p-adic
completion
of
K;
e
v
:
O
v
→
O
Ω
for
the
natural
evaluation
homomorphism
over
O
K
associated
to
x
v
.
Next,
observe
that,
since
O
v
⊗
O
K
K
is
a
valuation
ring
[contained
in
the
function
field
K(X
K
)
of
the
hyperbolic
curve
X
K
over
K]
that
contains
K,
and
whose
field
of
fractions
coincides
with
K(X
K
)
[cf.
Definition
2.2,
(ii)],
it
follows
immediately
from
the
well-known
theory
of
one-dimensional
function
fields
over
an
algebraically
closed
field
that
Ω
O
v
/O
K
⊗
O
K
K
is
a
rank
one
free
module
over
O
v
⊗
O
K
K.
In
particular,
there
exists
an
element
t
∈
O
v
such
that
e
v
(t)
∈
m
Ω
,
and
dt
is
a
free
generator
of
Ω
O
v
/O
K
⊗
O
K
K
over
O
v
⊗
O
K
K.
Next,
we
observe
that,
for
any
positive
integer
N
,
there
exists
an
element
a
N
∈
m
K
such
that
e
v
(t
−
a
N
)
=
e
v
(t)
−
a
N
∈
p
N
O
Ω
.
N
∈
O
v
.
On
the
other
hand,
it
Note
that
the
above
equation
implies
that
t−a
p
N
follows
immediately
from
the
definition
of
the
module
of
relative
differentials
that
N
dt
=
d(t
−
a
N
)
=
p
N
·
d(
t−a
)
∈
Ω
O
v
/O
K
.
p
N
In
particular,
we
conclude
that
dt
is
a
nonzero
p-divisible
element
of
Ω
O
v
/O
K
.
Thus,
since
dt
is
a
free
generator
of
Ω
O
v
/O
K
⊗
O
K
K
over
O
v
⊗
O
K
K,
the
dif-
ferential
boundedness
of
v
would
imply
that
arbitrary
negative
integral
powers
of
p
are
contained
in
O
v
,
i.e.,
in
contradiction
to
our
assumption
that
v
is
a
p-valuation.
Hence
we
conclude
that
v
is
differentially
unbounded,
as
desired.
Next,
we
verify
that,
if
x
v
is
of
type
2,
then
v
is
differentially
bounded.
Suppose
that
x
v
is
of
type
2.
Then
it
follows
immediately
from
Proposition
3.4,
(iii),
that
there
exist
a
finite
extension
field
L
of
K,
a
compactified
semistable
∼
model
X
of
X
L
over
O
L
,
and
a
generic
point
x
of
X
s
such
that
O
v
→
O
X
,x
⊗
O
L
O
K
.
In
particular,
to
verify
that
v
is
differentially
bounded,
it
suffices
to
verify
that
Ω
O
X
,x
/O
L
is
a
finitely
generated
O
X
,x
-module.
On
the
other
hand,
this
follows
immediately
from
the
fact
that
O
X
,x
is
essentially
of
finite
type
over
O
L
.
Next,
we
verify
that,
if
x
v
is
of
type
3,
then
v
is
differentially
bounded.
Suppose
that
x
v
is
of
type
3.
Then
it
follows
from
Proposition
3.4,
(iii),
that
the
VE-chain
associated
to
the
p-valuation
v
is
strongly
edge-like.
Thus,
it
follows
immediately
from
Proposition
2.3,
(iii),
(iv)
[cf.
also
Definition
3.2,
(v),
[the
final
portion
of]
(vii);
Proposition
3.4,
(ii)]
that
there
exist
•
an
ascending
chain
K
=
K
0
⊆
K
1
⊆
·
·
·
⊆
K
i
⊆
·
·
·
of
finite
extension
fields
of
K
contained
in
K
and
indexed
by
N
82
and,
for
each
i
∈
N,
•
a
compactified
semistable
model
X
i
with
split
reduction
of
X
K
i
over
O
K
i
,
•
a
dominant
morphism
φ
i+1
:
X
i+1
−→
X
i
×
O
Ki
O
K
i+1
over
O
K
i+1
that
induces
the
identity
automorphism
on
the
generic
fiber,
•
a
node
e
i
of
(X
i
)
s
satisfying
the
following
conditions:
•
For
each
i
∈
N,
the
projection
to
X
i
of
φ
i+1
(e
i+1
)
coincides
with
the
node
e
i
∈
(X
i
)
s
⊆
X
i
.
•
The
equality
O
v
=
lim
A
i
⊗
O
Ki
O
K
−→
i∈N
def
—
where,
for
each
i
∈
N,
we
write
A
i
=
O
X
i
,e
i
;
the
transition
map
is
the
homomorphism
A
i
⊗
O
Ki
O
K
→
A
i+1
⊗
O
Ki+1
O
K
induced
by
the
composite
X
i+1
→
X
i
[which
maps
e
i+1
→
e
i
]
of
φ
i+1
with
the
projection
to
X
i
—
holds
[cf.
Proposition
2.3,
(vi)].
def
For
each
i
∈
N,
write
S
i
=
Spec
O
K
i
;
S
i
log
for
the
log
scheme
determined
by
the
log
structure
on
S
i
associated
to
the
closed
point
of
S
i
;
X
i
log
for
the
log
scheme
over
S
i
log
determined
by
the
natural
log
structure
on
X
i
[i.e.,
the
multiplicative
monoid
of
sections
of
O
X
i
that
are
invertible
on
the
open
subscheme
of
X
i
determined
by
X
K
i
];
ω
X
log
/S
log
,e
i
for
the
stalk
at
e
i
of
the
sheaf
of
relative
i
i
logarithmic
differentials
associated
to
the
proper,
log
smooth
morphism
X
i
log
→
S
i
log
[cf.
the
subsection
in
Notations
and
Conventions
entitled
“Log
schemes”].
Then
it
follows
immediately
from
the
definitions
of
the
various
log
structures
involved
that
the
morphism
φ
i+1
:
X
i+1
→
X
i
×
O
Ki
O
K
i+1
=
X
i
×
S
i
S
i+1
extends
to
a
log
étale
morphism
of
log
schemes
log
log
−→
X
i
log
×
S
log
S
i+1
,
X
i+1
i
which
induces
a
natural
isomorphism
∼
ω
X
log
/S
log
,e
i
⊗
A
i
A
i+1
→
ω
X
log
/S
log
,e
i+1
i
i
i+1
i+1
of
A
i+1
-modules,
hence
a
natural
homomorphism
φ
:
Ω
O
v
/O
K
=
lim
Ω
A
i
⊗
O
K
O
K
/O
K
−→
ω
X
log
/S
log
,e
0
⊗
A
0
O
v
−→
0
0
i
i∈N
83
of
O
v
-modules.
Here,
we
note
that
φ
induces
a
natural
isomorphism
∼
Ω
O
v
/O
K
⊗
O
K
K
→
ω
X
log
/S
log
,e
0
⊗
A
0
O
v
⊗
O
K
K
0
0
of
free
O
v
⊗
O
K
K-modules
of
rank
1.
Thus,
since
ω
X
log
/S
log
,e
0
is
a
finitely
0
0
generated
A
0
-module,
we
conclude
that
v
is
differentially
bounded,
as
desired.
Finally,
we
verify
that,
if
x
v
is
of
type
4,
then
v
is
differentially
bounded.
Suppose
that
x
v
is
of
type
4.
Then
it
follows
from
Proposition
3.3,
(ii);
Propo-
sition
3.4,
(iii)
[cf.
also
Remark
3.1.1],
that
the
VE-chain
associated
to
the
p-valuation
v
[cf.
Proposition
2.3,
(viii)]
is
weakly
verticial.
Thus,
it
follows
immediately
from
Proposition
2.3,
(iii),
(iv)
[cf.
also
Definition
3.2,
(iii);
the
final
portion
of
Remark
2.1.4]
that
there
exist
•
an
ascending
chain
K
=
K
0
⊆
K
1
⊆
·
·
·
⊆
K
i
⊆
·
·
·
of
finite
extension
fields
of
K
contained
in
K
and
indexed
by
N
and,
for
each
i
∈
N,
•
a
compactified
semistable
model
X
i
with
split
reduction
of
X
K
i
over
O
K
i
,
•
a
dominant
morphism
φ
i+1
:
X
i+1
−→
X
i
×
O
Ki
O
K
i+1
over
O
K
i+1
that
induces
the
identity
automorphism
on
the
generic
fiber,
•
an
irreducible
component
v
i
of
(X
i
)
s
satisfying
the
following
conditions:
•
For
each
i
∈
N,
the
projection
to
X
i
of
φ
i+1
(v
i+1
)
is
a
closed
point
x
i
∈
(X
i
)
s
⊆
X
i
of
(X
i
)
s
that
lies
in
the
smooth
locus
of
v
i
.
•
For
each
i
∈
N,
there
exists
a
section
ψ
i
:
Spec
O
K
i
→
X
i
whose
image
contains
x
i
.
•
The
equality
O
v
=
lim
A
i
⊗
O
Ki
O
K
−→
i∈N
def
—
where,
for
each
i
∈
N,
we
write
A
i
=
O
X
i
,x
i
;
the
transition
map
is
the
homomorphism
A
i
⊗
O
Ki
O
K
→
A
i+1
⊗
O
Ki+1
O
K
induced
by
the
composite
X
i+1
→
X
i
[which
maps
x
i+1
→
x
i
]
of
φ
i+1
with
the
projection
to
X
i
—
holds
[cf.
Proposition
2.3,
(vi)].
84
Thus,
we
are
in
the
situation
of
Proposition
3.7,
(i).
In
particular,
there
exists
a
collection
{t
i
,
γ
i+1
,
π
i+1
}
i∈N
of
elements
t
i
∈
A
i
and
γ
i+1
,
π
i+1
∈
m
K
i+1
as
in
Proposition
3.7,
(i),
such
that
t
i
=
γ
i+1
+
π
i+1
t
i+1
.
Next,
observe
that
since
x
i
lies
in
the
smooth
locus
of
(X
i
)
s
,
it
follows
that
Ω
A
i
/O
Ki
is
a
free
A
i
-module
of
rank
1
generated
by
dt
i
.
In
particular,
since
t
i
=
γ
i+1
+
π
i+1
t
i+1
,
we
conclude
that
1
Ω
A
i+1
/O
Ki+1
=
·
Ω
A
i
/O
Ki
⊗
A
i
A
i+1
.
π
i+1
Thus,
it
follows
immediately
from
the
equality
O
v
=
lim
i∈N
A
i
⊗
O
Ki
O
K
that
−→
1
·
Ω
A
0
/O
K
0
⊗
A
0
O
v
.
Ω
O
v
/O
K
=
lim
−→
1≤j≤i+1
π
j
i∈N
Now
suppose
that
v
is
differentially
unbounded.
For
each
positive
integer
i,
def
write
l
i
=
v
p
(π
i
).
Then
since
v
is
differentially
unbounded,
we
conclude
that
the
equality
l
i
=
+∞
i≥1
holds.
Next,
we
observe
that
it
follows
immediately
from
Proposition
3.7,
(ii)
[cf.
also
the
equality
O
v
=
lim
i∈N
A
i
⊗
O
Ki
O
K
],
that
v
determines
a
closed
−→
point
x
Ω
of
X
Ω
such
that
the
valuation
ring
of
the
point-theoretic
p-valuation
on
K(X
K
)
associated
to
x
Ω
dominates
O
v
,
hence
coincides
with
O
v
.
Thus,
we
conclude
that
x
v
is
the
point
∈
X
an
determined
by
x
Ω
,
hence
that
x
v
is
of
type
1,
in
contradiction
to
our
assumption
that
x
v
is
of
type
4.
This
completes
the
proof
of
Proposition
3.8.
Proposition
3.9
(Characterization
of
points
of
type
1
via
geometric
Σ-tempered
decomposition
groups).
In
the
notation
of
Definition
3.2,
sup-
pose
that
p
∈
Σ.
Let
x
∈
X
an
be
an
element;
l
∈
Σ
\
{p};
D
x
a
decomposition
of
X
associated
to
group
in
the
geometric
Σ-tempered
fundamental
group
Δ
Σ-tp
X
x.
Then
the
following
hold:
(i)
Suppose
that
x
is
of
type
1.
Then
D
x
is
trivial.
(ii)
Suppose
that
x
is
of
type
4.
Then
there
exists
an
open
subgroup
of
D
x
that
admits
a
continuous
surjective
homomorphism
to
Z
p
.
In
particular,
D
x
is
nontrivial.
(iii)
Suppose
that
x
is
of
type
i
∈
{2,
3},
and
that
X
satisfies
Σ-RNS.
Then
there
exists
an
open
subgroup
of
D
x
that
admits
a
continuous
surjective
homomorphism
to
Z
l
.
In
particular,
D
x
is
nontrivial.
(iv)
Suppose
that
X
satisfies
Σ-RNS.
Then
x
is
of
type
1
if
and
only
if
D
x
is
trivial.
85
an
be
a
lifting
of
x.
First,
we
observe
that
assertion
(i)
follows
Proof.
Let
x
∈
X
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(iii).
Suppose
that
x
[or,
equivalently,
x
]
is
of
type
i
∈
{2,
3},
and
that
X
satisfies
Σ-RNS.
Then
it
follows
immediately
from
Proposition
3.4,
(iii);
Proposition
3.5,
(i),
that
there
exists
an
open
subgroup
of
D
x
that
admits
a
continuous
surjective
homomorphism
to
Z
l
.
This
completes
the
proof
of
assertion
(iii).
Assertion
(iv)
follows
immediately
from
assertions
(i),
(ii),
(iii)
[cf.
also
Remark
3.1.1].
Thus,
to
complete
the
proof
of
Proposition
3.9,
it
suffices
to
verify
assertion
(ii).
To
verify
assertion
(ii),
by
replacing
K
by
a
suitable
extension
field
of
K
contained
in
Ω,
we
may
assume
without
loss
of
generality
[cf.
Proposition
3.4,
(iii)]
that
the
residue
field
of
K
is
separably
closed.
Suppose
that
x
[or,
equivalently,
x
]
is
of
type
4,
and
that
no
open
subgroup
of
D
x
admits
a
continuous
surjective
homomorphism
to
Z
p
.
Write
v
for
the
primitive
p-valuation
on
K(
X)
associated
to
x
[cf.
Proposition
2.3,
(viii)];
v
for
the
p-valuation
obtained
by
restricting
v
to
K(X
K
).
Then
since
x
is
of
type
4,
it
follows
from
Proposition
3.8
that
Ω
O
v
/O
K
is
bounded.
In
particular,
by
replacing
K
by
a
finite
extension
field
of
K,
if
necessary,
we
observe
that
there
exist
•
a
positive
integer
N
and
•
a
compactified
semistable
model
with
split
reduction
X
of
X
over
O
K
such
that
the
center
z
on
X
of
the
VE-chain
associated
to
v
lies
in
the
smooth
locus
of
X
s
⊆
X
,
arises
from
a
point
of
X
valued
in
the
residue
field
of
O
K
,
and
satisfies
the
following
condition
[cf.
the
portion
of
the
proof
of
Proposition
3.8
concerning
points
of
type
4]:
(Ω
O
X
,z
/O
K
⊆)
Ω
O
X
,z
/O
K
⊗
O
X
,z
O
v
⊆
Ω
O
v
/O
K
⊆
1
·
Ω
O
X
,z
/O
K
⊗
O
X
,z
O
v
p
N
—
where
by
a
slight
abuse
of
notation,
we
use
the
notation
“⊆”
to
denote
the
various
natural
inclusions,
and
we
note
that
since
Ω
O
X
,z
/O
K
is
a
free
O
X
,z
-
module
of
rank
1,
the
O
v
-module
Ω
O
X
,z
/O
K
⊗
O
X
,z
O
v
is
a
free
O
v
-module
of
rank
1.
In
particular,
it
follows
immediately
that
the
second
and
third
inclusions
of
the
above
display
induce
the
injections
on
the
respective
p-adic
completions
[cf.
the
discussion
of
Remark
2.2.4].
In
the
remainder
of
the
proof
of
assertion
(ii),
we
suppose
that
we
are
in
the
situation
of
Proposition
2.12.
Moreover,
by
replacing
X
by
a
suitable
geo-
metrically
pro-Σ
connected
finite
étale
covering
of
X
[cf.
Proposition
2.3,
(xii);
Definition
2.7],
we
may
assume
without
loss
of
generality
that
X
=
Y
n
,
X
=
Y
n
,
z
=
(y
n
)
s
,
X
,z
=
O
Y
,y
,
O
n
n
X
,z
denotes
the
completion
of
the
local
ring
O
X
,z
.
Write
where
O
m
)
−→
O
×
Ψ
:
Hom
O
K
(
T
qtr,n
,
G
X
,z
m
)
for
the
assignment
discussed
in
Proposition
2.13,
(ii).
Let
f
∈
Hom
O
K
(
T
qtr,n
,
G
be
a
nontrivial
element
[which
exists
by
Propositions
2.11,
(i);
2.13,
(i)].
Thus,
86
the
logarithmic
differential
def
def
)
θ
=
dΨ(f
X
,z
/O
K
=
lim
Ω
(O
X
,z
/m
m
Ψ(f
)
∈
Ω
O
z
)/O
K
←−
m≥1
—
where
m
z
denotes
the
maximal
ideal
of
O
X
,z
;
m
ranges
over
the
positive
×
is
=
0
[cf.
the
first
display,
as
well
as
the
discussion
integers
—
of
Ψ(f
)
∈
O
X
,z
following
this
first
display,
in
the
proof
of
Theorem
2.16,
where
we
observe
that
this
portion
of
the
proof
of
Theorem
2.16
may
be
applied
even
in
the
case
of
the
“K”
—
i.e.,
with
separably
closed
residue
field
—
of
the
present
discussion].
v
for
the
field
of
fractions
of
v
for
the
p-adic
completion
of
O
v
;
K
Next,
write
O
v
.
Since
v
is
a
real
valuation
[cf.
Proposition
2.3,
(vii);
Remark
3.1.1],
it
follows
O
immediately
from
the
final
portion
of
Remark
2.2.4
that
the
henselization
of
O
v
v
.
Write
H
⊆
Δ
Σ-tp
for
the
closed
subgroup
may
be
regarded
as
a
subring
of
O
X
obtained
by
forming
the
intersection
of
the
kernels
of
the
continuous
surjective
H
,
K(
X)
D
x
⊆
K(
X)
for
the
subfields
fixed
homomorphisms
Δ
Σ-tp
Z
p
;
K(
X)
X
by
H
and
D
x
,
respectively.
Then
since
there
does
not
exist
any
continuous
surjective
homomorphism
D
x
Z
p
,
we
thus
conclude
that
D
x
⊆
H,
hence
that
H
⊆
K(
X)
D
x
⊆
K
v
.
K(
X)
On
the
other
hand,
it
follows
immediately
from
the
definition
of
the
center
z
on
X
that
there
exists
a
natural
homomorphism
φ
:
O
X
,z
→
O
v
of
local
rings,
X
,z
→
O
v
of
topological
local
rings.
which
thus
induces
a
homomorphism
φ
:
O
def
=
0.
Now
we
claim
that
φ
is
injective.
Indeed,
suppose
that
p
=
Ker(
φ)
Then
since
O
X
,z
is
a
regular
local
ring
of
dimension
2,
and
O
v
is
p-torsion-free,
it
follows
that
p
is
a
prime
ideal
of
height
1
such
that
p
∩
O
K
=
{0}.
Next,
X
,z
/p)⊗
O
(O
K
/m
K
)
is
finite
over
O
K
/m
K
.
Thus,
since
O
X
,z
/p
observe
that
(
O
K
is
a
complete
O
K
-module,
we
conclude
that
O
X
,z
/p
is
finite
over
O
K
.
On
the
v
is
other
hand,
observe
that
the
composite
homomorphism
O
X
,z
→
O
v
→
O
X
,z
/p
is
injective.
injective,
hence
that
the
natural
homomorphism
O
X
,z
→
O
In
particular,
we
conclude
that
O
X
,z
⊗
O
K
K
embeds
into
a
finite
dimensional
K-vector
space,
a
contradiction.
This
completes
the
proof
of
our
claim
that
φ
is
injective.
Next,
we
observe
that
since
the
image
of
Ψ(f
)
is
p-divisible
in
the
multi-
X
,z
⊗
O
H
}
×
[cf.
Proposition
2.13,
(ii)],
the
image
plicative
group
{
O
K(
X)
X
,z
×
,
hence
also
in
the
multiplicative
group
of
Ψ(f
)
in
the
multiplicative
group
K
v
×
O
v
,
is
p-divisible.
Write
Ω
O
v
/O
def
K
=
lim
Ω
(O
v
/p
m
·O
v
)/O
K
,
←−
m≥1
where
m
ranges
over
the
positive
integers.
Then
it
follows
immediately
from
well-known
basic
facts
concerning
modules
of
differentials,
together
with
the
fact
that
Ω
O
X
,z
/O
K
is
a
free
O
X
,z
-module
of
rank
1,
that
Ω
O
X
,z
/O
K
=
lim
Ω
O
X
,z
/O
K
⊗
O
X
,z
O
X
,z
/m
m
z
=
Ω
O
X
,z
/O
K
⊗
O
X
,z
O
X
,z
;
←−
m≥1
87
Ω
O
v
/O
K
=
lim
Ω
O
v
/O
K
⊗
Z
Z/p
m
Z.
←−
m≥1
In
particular,
since
φ
is
injective,
it
thus
follows
from
the
discussion
of
the
final
portion
of
the
second
paragraph
of
the
present
proof
that
X
,z
⊆
Ω
O
/O
⊗
O
O
v
⊆
Ω
Ω
O
X
,z
/O
K
=
Ω
O
X
,z
/O
K
⊗
O
X
,z
O
X
,z
X
,z
K
O
v
/O
,
K
hence
that
the
image
θ
v
∈
Ω
O
v
/O
of
θ
in
Ω
O
v
/O
is
=
0.
On
the
other
hand,
K
K
×
is
p-divisible,
and
Ω
since
the
image
of
Ψ(f
)
in
O
is,
by
definition,
p-
O
v
/O
K
v
adically
separated,
we
conclude
that
θ
v
=
0,
a
contradiction.
This
completes
the
proof
of
assertion
(ii),
hence
of
Proposition
3.9.
Corollary
3.10
(Reconstruction
of
points
of
type
1
via
geometric
tem-
pered
fundamental
groups).
Let
Σ
⊆
Primes
be
a
subset
of
cardinality
≥
2
that
contains
p;
K
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
hyperbolic
curve
over
K.
Write
Ω
for
the
p-adic
completion
of
K;
Π
tp
(−)
for
the
Σ-tempered
fundamental
group
of
(−);
X
→
X
tp
for
the
universal
pro-Σ
covering
corresponding
to
Π
[so
Gal(
X/X)
may
be
X
identified
with
the
pro-Σ
completion
of
Π
tp
X
].
Suppose
that
X
satisfies
Σ-RNS.
Then
the
set
X(Ω)
equipped
with
its
natural
action
by
Gal(
X/X)
—
hence
also,
the
quotient
set
X(Ω)
X(Ω)
—
by
passing
to
the
set
of
Gal(
X/X)-orbits,
may
be
reconstructed,
in
a
purely
combinatorial/group-theoretic
way
and
func-
torially
with
respect
to
isomorphisms
of
topological
groups,
from
the
underlying
topological
group
of
Π
tp
X
.
Proof.
Recall
that,
for
any
hyperbolic
curve
Y
over
K,
the
set
of
cuspidal
inertia
subgroups
of
Π
tp
Y
,
hence
also
the
genus
of
Y
,
may
be
reconstructed,
in
a
purely
combinatorial/group-theoretic
way
and
functorially
with
respect
to
isomorphisms
of
topological
groups,
from
the
underlying
topological
group
of
Π
tp
Y
[cf.
the
generalized
version
of
[SemiAn],
Corollary
3.11,
discussed
in
[AbsTopII],
Remark
2.11.1,
(i)].
On
the
other
hand,
in
the
case
where
X
is
a
proper
hyperbolic
curve
over
K,
we
observe
that
Corollary
3.10
follows
immediately
from
Proposition
3.9,
(iv),
and
[the
proof
of]
Corollary
2.5,
(i).
Thus,
by
applying
this
observation
to
the
Σ-tempered
fundamental
groups
of
the
smooth
compactifications
of
the
various
[connected]
geometrically
pro-Σ
finite
étale
Galois
coverings
of
X
over
K
of
genus
≥
2,
we
conclude
that
X(Ω)
equipped
with
its
natural
action
by
Gal(
X/X)
may
be
reconstructed,
in
a
purely
combinatorial/group-theoretic
way
and
functorially
with
respect
to
isomorphisms
of
topological
groups,
from
the
underlying
topological
group
of
Π
tp
X
.
This
completes
the
proof
of
Corollary
3.10.
88
Theorem
3.11
(Preservation
of
decomposition
subgroups
associated
to
closed
points).
For
∈
{†,
‡},
let
p
be
a
prime
number;
Σ
⊆
Primes
a
subset
that
contains
p
;
l
∈
(Σ
†
\{p
†
})
∩
(Σ
‡
\{p
‡
});
K
a
mixed
characteristic
complete
discrete
valuation
fields
of
residue
characteristic
p
;
X
a
hyperbolic
curve
over
K
;
L
⊆
K
a
tamely
ramified
[not
necessarily
finite!]
Galois
extension
of
K
that
may
be
written
as
a
union
of
finite
tamely
ramified
Galois
extensions
of
K
in
K
of
ramification
index
prime
to
l.
Let
σ
:Π
(Σ
†
)
∼
(Σ
‡
)
→
Π
‡
†
X
†
X
‡
L
L
be
an
isomorphism
of
profinite
groups
between
the
geometrically
pro-Σ
†
étale
fundamental
group
of
X
L
†
†
and
the
geometrically
pro-Σ
‡
étale
fundamental
group
of
X
L
‡
†
.
For
∈
{†,
‡},
write
Δ
Σ
for
the
geometric
pro-Σ
étale
fundamental
X
L
group
of
X
L
,
I
L
⊆
G
L
for
the
inertia
subgroup
of
G
L
,
and
k
L
for
the
residue
field
of
L
.
Then
the
following
hold:
(i)
We
have
an
equality
p
†
=
p
‡
,
and
σ
induces
isomorphisms
of
profinite
†
‡
∼
∼
→
Δ
Σ
,
G
L
†
→
G
L
‡
.
In
particular,
Σ
†
=
Σ
‡
.
Finally,
if,
groups
Δ
Σ
X
†
X
‡
L
†
L
‡
for
each
∈
{†,
‡},
every
pro-l
closed
subgroup
of
the
kernel
of
the
l-adic
cyclotomic
character
on
G
k
L
is
trivial
[cf.
Remark
3.11.1
below],
then,
for
all
sufficiently
small
open
subgroups
J
†
⊆
G
L
†
,
J
‡
⊆
G
L
‡
such
that
∼
σ
induces
an
isomorphism
J
†
→
J
‡
,
σ
also
induces
an
isomorphism
of
profinite
groups
between
the
respective
images
of
J
†
∩
I
L
†
,
J
‡
∩
I
L
‡
in
the
maximal
pro-l
quotients
J
†
(J
†
)
{l}
,
J
‡
(J
‡
)
{l}
.
(ii)
Suppose
that,
for
all
sufficiently
small
open
subgroups
J
†
⊆
G
L
†
,
J
‡
⊆
∼
G
L
‡
such
that
σ
induces
an
isomorphism
J
†
→
J
‡
,
σ
also
induces
an
isomorphism
of
profinite
groups
between
the
respective
images
of
J
†
∩
I
L
†
,
J
‡
∩
I
L
‡
in
the
maximal
pro-l
quotients
J
†
(J
†
)
{l}
,
J
‡
(J
‡
)
{l}
.
def
Write
Σ
=
Σ
†
=
Σ
‡
[cf.
(i)].
Suppose,
moreover,
that
X
†
and
X
‡
satisfy
Σ-RNS.
Then
σ
induces
a
bijection
between
the
respective
sets
of
decomposition
subgroups
associated
to
closed
points
of
X
L
†
†
and
X
L
‡
‡
,
where
‡
denote
the
respective
completions
of
L
†
,
L
‡
.
†
,
L
L
def
def
Proof.
First,
we
verify
assertion
(i).
Write
τ
†
=
σ
−1
,
τ
‡
=
σ.
For
∈
{†,
‡},
write
for
the
unique
element
of
{†,
‡}
\
{}.
Then
observe
that
it
follows
immediately,
by
applying
to
τ
(Δ
Σ
),
for
both
=
†
and
=
‡,
X
L
•
the
argument
of
the
proof
of
[MiTs1],
Corollary
4.6
[in
the
case
where
the
extension
L
/K
is
finite;
here,
we
note
that
in
this
case,
it
follows
from
(Σ
)
[MiSaTs],
Theorem
3.8,
that
if
Π
X
is
topologically
finitely
generated,
L
then
the
extension
L
/K
is
also
finite],
and
89
•
[MiSaTs],
Theorem
3.8
[in
the
case
where
the
extension
L
/K
is
infinite],
that
σ
induces
isomorphisms
of
profinite
groups
†
∼
‡
Σ
Δ
Σ
X
†
→
Δ
X
‡
;
∼
G
L
†
→
G
L
‡
.
Thus,
we
conclude
from
[MiTs2],
Theorem
A,
(i),
together
with
the
well-known
structure
of
geometric
fundamental
groups
of
hyperbolic
curves
over
fields
of
characteristic
zero
[cf.,
e.g.,
[MT],
Remark
1.2.2],
that
p
†
=
p
‡
,
and
Σ
†
=
Σ
‡
.
The
final
portion
of
assertion
(i)
follows
immediately
from
the
well-known
structure,
for
∈
{†,
‡},
of
the
Galois
group
Gal((K
)
tm
/K
)
over
K
of
the
maximal
tamely
ramified
extension
(K
)
tm
of
K
[under
the
assumption
that
every
pro-l
closed
subgroup
of
the
kernel
of
the
l-adic
cyclotomic
character
on
G
k
L
is
trivial],
which
implies
that,
for
any
sufficiently
small
open
subgroup
J
⊆
G
L
,
the
image
of
J
∩
I
L
in
the
maximal
pro-l
quotient
J
(J
)
{l}
coincides
with
the
unique
maximal
abelian
normal
closed
subgroup
of
(J
)
{l}
.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
we
note
that
it
follows
from
assertion
def
(i)
that
p
=
p
†
=
p
‡
.
Thus,
in
light
of
our
assumption
on
σ
[cf.
also
assertion
(i)],
it
follows
from
[CmbGC],
Corollary
2.7,
(i)
[applied
in
the
case
where
“l”
is
taken
to
be
the
p
of
the
present
discussion],
(iii)
[applied
in
the
case
where
“l”
is
∼
taken
to
be
the
l
of
the
present
discussion],
that
the
isomorphism
Δ
Σ
→
Δ
Σ
X
†
X
‡
L
†
L
†
[cf.
(i)]
satisfies
the
condition
(b
∃
)
of
[CbTpIII],
Proposition
3.6.
In
particular,
by
applying
[CbTpIII],
Proposition
3.6,
(i),
we
conclude
that
the
isomorphism
∼
Δ
Σ
→
Δ
Σ
arises,
up
to
composition
with
an
inner
automorphism,
from
an
X
†
X
‡
L
†
L
‡
isomorphism
between
the
respective
geometric
Σ-tempered
fundamental
groups
of
X
†
and
X
‡
.
Thus,
by
replacing
σ
by
the
composite
of
σ
with
an
inner
automorphism
arising
from
Δ
Σ
,
we
may
assume
without
loss
of
generality
X
‡
L
‡
that
σ
arises
from
an
isomorphism
between
the
respective
pull-backs
via
the
natural
inclusions
G
L
†
⊆
G
K
†
,
G
L
‡
⊆
G
K
‡
of
the
geometrically
Σ-tempered
fundamental
groups
of
X
†
and
X
‡
.
†
,
X
‡
for
the
universal
geometrically
pro-Σ
coverings
corre-
Next,
write
X
(Σ)
(Σ)
sponding
to
Π
†
,
Π
‡
,
respectively;
Ω
†
,
Ω
‡
for
the
p-adic
completions
of
†
‡
X
†
L
X
‡
L
K
,
K
,
respectively.
Then
since
σ
determines
an
isomorphism
between
the
respective
geometric
Σ-tempered
fundamental
groups
of
X
†
and
X
‡
,
it
follows
immediately
from
Corollary
3.10
that
σ
induces
a
bijection
∼
‡
†
(Ω
†
)
→
X
(Ω
‡
)
X
that
is
compatible
with
the
respective
natural
actions
of
Π
(Σ)
(Σ)
,
Π
‡
.
X
†
†
X
†
L
L
Thus,
in
light
of
[Tate],
§3.3,
Theorem
1
[which,
as
is
easily
verified,
admits
a
routine
‡
,
†
,
L
generalization
to
mixed
characteristic
complete
valuation
fields
such
as
L
i.e.,
whose
valuations
are
not
necessarily
discrete
[but
nonetheless
tamely
ram-
ified
over
some
discrete
valuation],
and
whose
residue
fields
are
not
necessarily
90
perfect],
we
conclude
that
σ
induces
a
bijection
between
the
respective
sets
of
decomposition
subgroups
associated
to
closed
points
of
X
L
†
†
and
X
L
‡
‡
.
This
completes
the
proof
of
assertion
(ii),
hence
of
Theorem
3.11.
Remark
3.11.1.
In
passing,
we
observe
that
the
condition
concerning
the
kernel
of
the
l-adic
cyclotomic
character
on
G
k
L
that
appears
in
the
final
portion
of
Theorem
3.11,
(i),
is
satisfied
if
k
L
is
either
separably
closed
or
algebraic
over
the
finite
field
of
cardinality
p.
We
are
now
in
a
position
to
verify
an
absolute
version
of
the
Grothendieck
Conjecture
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields
[cf.
Theorem
3.12
below],
which
is
one
of
the
central
open
questions
in
anabelian
geometry.
Theorem
3.12
(Absolute
version
of
the
Grothendieck
Conjecture
for
arbitrary
hyperbolic
curves
over
p-adic
local
fields).
Let
p
†
,
p
‡
be
prime
numbers;
Σ
⊆
Primes
a
subset
of
cardinality
≥
2
that
contains
p
†
and
p
‡
;
K
†
,
K
‡
mixed
characteristic
local
fields
of
residue
characteristic
p
†
,
p
‡
,
respectively;
X
†
,
X
‡
hyperbolic
curves
over
K
†
,
K
‡
,
respectively.
Then
the
natural
map
Isom(X
†
,
X
‡
)
−→
OutIsom(Π
X
†
,
Π
X
‡
)
(Σ)
(Σ)
is
bijective.
Proof.
First,
we
observe
that
any
isomorphism
of
schemes
between
X
†
and
X
‡
necessarily
lies
over
an
isomorphism
of
fields
between
K
†
and
K
‡
.
[Indeed,
this
×
follows
immediately
by
considering
subgroups
of
the
groups
of
units
Γ(X
†
,
O
X
†
),
×
‡
Γ(X
,
O
X
‡
)
whose
unions
with
{0}
are
closed
under
addition.]
Now
Theorem
3.12
follows
immediately
by
combining
Theorems
2.17;
3.11,
(i),
(ii)
[cf.
also
Remark
3.11.1],
of
the
present
paper
with
[AbsTopII],
Corollary
2.9.
Remark
3.12.1.
Theorem
3.12
may
be
regarded
as
a
complete
affirmative
resolu-
tion
of
the
absolute
version
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
over
p-adic
local
fields
in
the
geometrically
pro-Σ
case,
where
Σ
⊆
Primes
is
a
subset
of
cardinality
≥
2
that
contains
the
residue
characteristic
of
the
base
field.
On
the
other
hand,
the
following
questions
remain
open,
to
the
authors’
knowledge,
at
the
time
of
writing
of
the
present
paper:
Question
1:
Can
one
prove
a
geometrically
pro-p
version
of
the
abso-
lute
Grothendieck
Conjecture
for
hyperbolic
curves
over
p-adic
local
fields?
In
this
context,
we
observe
that
certain
partial
results
in
this
direction
are
obtained
in
[Hgsh].
Question
2:
Can
one
prove
an
absolute
version
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
over
more
general
base
fields?
For
instance,
one
may
consider
the
case
where
the
base
fields
are
mixed
91
characteristic
complete
discrete
valuation
fields
whose
residue
fields
are
algebraic
over
F
p
,
i.e.,
a
class
of
fields
for
which
a
relative
ver-
sion
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
has
been
known
for
some
time
[cf.
[AnabTop],
Theorem
4.12].
Theorem
3.13
(Absolute
version
of
the
Grothendieck
Conjecture
for
configuration
spaces
associated
to
arbitrary
hyperbolic
curves
over
p-adic
local
fields).
Let
p
†
,
p
‡
be
prime
numbers;
K
†
,
K
‡
mixed
characteristic
local
fields
of
residue
characteristic
p
†
,
p
‡
,
respectively;
X
†
,
X
‡
hyperbolic
curves
over
K
†
,
K
‡
,
respectively;
n
†
,
n
‡
positive
integers.
Write
X
n
†
†
(respectively,
X
n
‡
‡
)
for
the
n
†
-th
(respectively,
n
‡
-th)
configuration
space
associated
to
X
†
(respectively,
X
‡
).
Then
the
natural
map
Isom(X
n
†
†
,
X
n
‡
‡
)
−→
OutIsom(Π
X
†
,
Π
X
‡
)
n
†
n
‡
is
bijective.
Proof.
First,
we
observe
that
any
isomorphism
of
schemes
between
X
n
†
†
and
X
n
‡
‡
necessarily
lies
over
an
isomorphism
of
fields
between
K
†
and
K
‡
.
[Indeed,
this
follows
immediately
by
a
similar
argument
to
the
argument
applied
in
the
proof
of
Theorem
3.12.]
Now
Theorem
3.13
follows
immediately
from
a
def
routine
argument
via
induction
on
n
=
n
†
=
n
‡
[cf.
[AbsTopI],
Theorem
2.6,
(v);
[HMM],
Theorem
A,
(i),
(ii)],
by
combining
Theorem
3.12
of
the
present
paper
with
the
relative
version
of
the
Grothendieck
Conjecture
given
in
[LocAn],
Theorem
A.
Next,
we
discuss
the
functorial
behavior
of
the
lengths
of
nodes
of
special
fibers
of
compactified
semistable
models
[cf.
Definition
3.14
below]
with
respect
to
finite
morphisms
between
compactified
semistable
models
that
extend
finite
étale
Galois
coverings
of
hyperbolic
curves
over
mixed
characteristic
complete
discrete
valuation
fields.
Definition
3.14.
Let
K
be
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
hyperbolic
curve
over
K;
X
a
compactified
semistable
model
with
split
reduction
of
X
over
O
K
;
e
a
node
of
X
s
.
Recall
that
the
completion
of
the
local
ring
O
X
,e
at
e
is
isomorphic
to
O
K
[[x,
y]]/(xy
−
a),
where
x,
y
denote
indeterminates;
a
∈
m
K
\
{0}.
Then
we
shall
refer
to
v
p
(a)
as
the
length
of
e.
[Note
that
the
length
of
e
is
independent
of
the
choice
of
a,
∼
as
well
as
of
the
isomorphism
O
X
,e
→
O
K
[[x,
y]]/(xy
−
a)
over
O
K
[cf.
[Hur],
§3.7].]
92
Proposition
3.15
(Functorial
behavior
of
the
lengths
of
nodes).
Let
K
be
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
character-
istic
p;
X
a
hyperbolic
curve
over
K;
Y
→
X
a
[connected]
finite
étale
Galois
covering
of
hyperbolic
curves
over
K;
Y
a
compactified
semistable
model
of
Y
def
over
O
K
that
is
stabilized
by
G
=
Gal(Y
/X).
Write
X
for
the
compactified
semistable
model
of
X
over
O
K
obtained
by
forming
the
quotient
of
Y
by
the
action
of
G
on
Y
[cf.
Proposition
2.3,
(iv)];
f
:
Y
→
X
for
the
natural
quotient
morphism.
Suppose
that
•
Y
has
split
reduction,
and
that
•
the
natural
action
of
G
on
Y
s
does
not
permute
the
branches
of
some
node
e
Y
of
Y
s
.
def
Write
e
X
=
f
(e
Y
)
for
the
node
of
X
s
determined
by
e
Y
[cf.
Proposition
2.3,
(iv)];
l
X
,
l
Y
for
the
lengths
of
the
nodes
e
X
,
e
Y
[relative
to
the
compactified
semistable
models
X
,
Y,
respectively].
Then
there
exists
a
positive
integer
m
such
that
l
X
=
m
·
l
Y
.
Moreover,
the
positive
integer
m
may
be
computed
as
the
cardinality
of
the
decomposition
subgroup
[i.e.,
the
stabilizer
subgroup]
of
e
Y
in
G.
def
Proof.
Write
S
=
Spec
O
K
;
S
log
for
the
log
scheme
obtained
by
equipping
S
with
the
log
structure
determined
by
the
closed
point
of
S;
X
log
,
Y
log
for
the
log
schemes
over
S
log
determined
by
the
compactified
semistable
models
X
,
Y,
respectively
[cf.
the
discussion
of
the
subsection
in
Notations
and
Conventions
entitled
“Log
schemes”].
Observe
that
f
naturally
determines
a
finite
morphism
f
log
:
Y
log
→
X
log
of
log
schemes,
hence
a
finite
morphism
X
,e
)
log
,
Y,e
)
log
−→
(Spec
O
(Spec
O
Y
X
Y,e
denote
the
completions
of
the
respective
normal
local
rings
X
,e
,
O
where
O
X
Y
O
X
,e
X
,
O
Y,e
Y
,
and
the
superscripts
“log”
denote
the
log
structures
induced
by
the
respective
log
structures
of
X
log
,
Y
log
.
Since
Y
is
assumed
to
have
split
reduction,
it
follows
[cf.
the
discussion
of
Definition
3.14]
that
X
,e
∼
O
K
[[u
1
,
u
2
]]/(u
1
u
2
−
a),
O
X
=
Y,e
∼
O
K
[[v
1
,
v
2
]]/(v
1
v
2
−
b),
O
Y
=
where
u
1
,
u
2
,
v
1
,
v
2
denote
indeterminates;
a,
b
∈
m
K
\{0}
are
elements
such
that
l
X
=
v
p
(a),
l
Y
=
v
p
(b).
Moreover,
it
follows
immediately
from
the
definitions
of
the
log
structures
involved,
together
with
the
geometry
of
the
irreducible
X
,e
and
Spec
O
Y,e
,
that,
after
components
of
the
special
fibers
of
Spec
O
X
Y
possibly
switching
the
indices
∈
{1,
2}
of
[either
or
both
of]
the
pairs
(u
1
,
u
2
)
×
and
(v
1
,
v
2
),
there
exist
positive
integers
m
1
,
m
2
and
units
c
1
,
c
2
∈
O
Y,e
Y
such
that
m
1
≥
m
2
,
and
u
1
=
c
1
·
v
1
m
1
,
u
2
=
c
2
·
v
2
m
2
,
93
Y,e
via
the
natural
injection
O
X
,e
→
where
we
regard
u
1
,
u
2
as
elements
in
O
Y
X
O
Y,e
Y
.
In
particular,
it
holds
that
a
=
u
1
u
2
=
c
1
c
2
v
1
m
1
v
2
m
2
=
c
1
c
2
v
1
m
1
−m
2
b
m
2
∈
O
K
[[v
1
,
v
2
]]/(v
1
v
2
−
b).
On
the
other
hand,
such
a
relation
implies,
in
light
of
the
well-known
structure
of
the
log
structures
involved,
i.e.,
in
effect,
the
geometry
of
the
irreducible
X
,e
and
Spec
O
Y,e
,
that
m
def
components
of
the
special
fibers
of
Spec
O
=
X
Y
m
m
1
=
m
2
,
hence
that
a
=
c
1
c
2
b
.
In
particular,
we
conclude
that
l
X
=
m
·
l
Y
,
as
desired.
Finally,
the
fact
that
m
may
be
computed
as
the
cardinality
of
the
decomposition
subgroup
[i.e.,
the
stabilizer
subgroup]
of
e
Y
in
G
follows
immediately
from
the
fact
that
the
generic
degree
of
the
[finite,
generically
Y,e
→
Spec
O
X
,e
is
[easily
computed,
via
the
explicit
étale]
morphism
Spec
O
Y
X
presentations
of
O
Y,e
Y
,
O
X
,e
X
given
above,
to
be]
m.
This
completes
the
proof
of
Proposition
3.15.
Next,
we
apply
Proposition
3.15,
together
with
the
theory
of
p-adic
arith-
metic
cuspidalizations
developed
in
[Tsjm],
§2,
to
prove
that
the
various
p-adic
versions
of
the
Grothendieck-Teichmüller
group
that
appear
in
the
literature
[cf.
[Tsjm],
Remark
2.1.2]
in
fact
coincide.
Theorem
3.16
(Equality
of
various
p-adic
versions
of
the
Grothendieck-
def
-Teichmüller
group).
Write
X
=
P
1
Q
\
{0,
1,
∞};
p
GT
⊆
Out(Π
X
)
for
the
Grothendieck-Teichmüller
group
[cf.
[CmbCsp],
Remark
1.11.1];
GT
M
⊆
GT
(⊆
Out(Π
X
))
for
the
metrized
Grothendieck-Teichmüller
group
[cf.
[CbTpIII],
Remark
3.19.2];
def
GT
tp
=
GT
∩
Out(Π
tp
p
X
)
⊆
Out(Π
X
)
[cf.
the
subsection
in
Notations
and
Conventions
entitled
“Fundamental
groups”;
[Tsjm],
Definition
2.1].
Then
the
natural
inclusion
GT
M
⊆
GT
tp
p
of
subgroups
of
GT
is
an
equality.
In
particular,
it
holds
that
GT
M
=
GT
p
=
GT
G
=
GT
tp
p
[cf.
[Tsjm],
Remark
2.1.2].
94
Proof.
First,
we
recall
that
there
exists
a
natural
surjection
φ
:
GT
tp
p
G
Q
p
whose
restriction
to
G
Q
p
is
the
identity
automorphism
[cf.
[Tsjm],
Corollary
B,
as
well
as
Remark
3.16.1
below].
Thus,
since
G
Q
p
⊆
GT
M
⊆
GT
tp
p
,
it
suffices
to
prove
that
Ker(φ)
⊆
GT
M
.
Let
σ
∈
Ker(φ).
Fix
a
lifting
σ̃
∈
Aut(Π
tp
X
)
of
will
always
denote
the
Primes-
σ.
[Here
and
in
the
following
discussion,
Π
tp
(−)
tempered
fundamental
group
of
(−).]
Then
it
follows
immediately
from
the
construction
of
φ
[cf.
the
discussion,
in
the
proof
of
[Tsjm],
Corollary
2.4,
of
the
two
paragraphs
following
the
proof
of
Claim
2.4.B;
the
discussion,
in
the
proof
of
[HMT],
Theorem
4.4,
of
the
observa-
tion
immediately
following
the
statement
of
Claim
4.4.A;
[NCBel],
Corollary
1.2]
that,
for
any
finite
subset
S
⊆
Q
\
{0,
1}
⊆
X(Q
p
),
σ̃
lifts
to
an
automorphism
def
of
Π
tp
X
S
[where
we
write
X
S
=
X
\
S]
with
respect
to
the
natural
surjection
tp
Π
tp
X
S
Π
X
determined
[up
to
composition
with
an
inner
automorphism]
by
the
natural
open
immersion
X
S
→
X.
Next,
let
ψ
Y
:
Y
→
X
be
a
connected
finite
étale
covering
over
Q
p
.
Write
ψ
Z
:
Z
→
X
for
the
Galois
closure
of
ψ
Y
;
Z
for
the
compactified
stable
model
of
σ
Z
over
O
Q
p
;
ψ
Y
σ
:
Y
σ
→
X,
ψ
Z
:
Z
σ
→
X
for
the
connected
finite
étale
coverings
tp
tp
tp
over
Q
p
that
correspond
to
the
open
subgroups
σ̃(Π
tp
Y
)
⊆
Π
X
,
σ̃(Π
Z
)
⊆
Π
X
,
respectively.
For
each
finite
subset
S
⊆
Q
\
{0,
1}
⊆
X(Q
p
),
write
def
•
Z
S
(respectively,
Z
S
σ
)
for
the
compactified
stable
model
of
Z
S
=
Z
\
−1
σ
−1
(S)
(respectively,
Z
S
σ
=
Z
σ
\
(ψ
Z
)
(S))
over
O
Q
p
;
ψ
Z
def
•
X
S
(respectively,
Y
S
,
Y
S
σ
)
for
the
compactified
semistable
model
of
X
S
(respectively,
Y
S
=
Y
\
ψ
Y
−1
(S),
Y
S
σ
=
Y
σ
\
(ψ
Y
σ
)
−1
(S))
obtained
by
forming
the
quotient
of
Z
S
(respectively,
Z
S
,
Z
S
σ
)
via
the
natural
action
of
Gal(Z/X)
(respectively,
Gal(Z/Y
),
Gal(Z
σ
/Y
σ
))
[cf.
Proposition
2.3,
(iv)].
def
def
Next,
observe
that
there
exists
a
finite
subset
T
⊆
Q
\
{0,
1}
⊆
X(Q
p
)
such
that
•
the
natural
action
of
Gal(Z/X)
on
(Z
T
)
s
does
not
permute
any
branches
of
nodes,
and
•
X
T
,
Y
T
,
Y
T
σ
are
the
respective
compactified
stable
models
of
X
T
,
Y
T
,
Y
T
σ
over
O
Q
p
.
Then
since
σ̃
lifts
to
an
automorphism
of
Π
tp
X
T
[cf.
the
above
discussion],
hence
to
an
isomorphism
∼
tp
tp
tp
tp
tp
Π
tp
Z
×
Π
tp
Π
X
T
=
Π
Z
T
→
Π
Z
σ
=
Π
Z
σ
×
Π
tp
Π
X
T
,
T
X
95
X
it
follows
immediately
from
Proposition
2.3,
(iv),
together
with
[SemiAn],
Corol-
lary
3.11,
that
σ̃
induces
a
commutative
diagram
of
semi-graphs
∼
Γ
Z
T
−−−−→
Γ
Z
T
σ
⏐
⏐
⏐
⏐
Γ
X
T
Γ
X
T
,
where
Γ
(−)
denotes
the
dual
semi-graph
associated
to
(−)
s
,
compatible
with
the
respective
natural
actions
of
Gal(Z/X),
Gal(Z
σ
/X).
Thus,
we
conclude
∼
from
Proposition
3.15
that
the
isomorphism
Γ
Z
T
→
Γ
Z
T
σ
of
dual
semi-graphs
is
compatible
with
the
respective
metric
structures
[cf.
[CbTpIII],
Definition
3.5,
(iii)].
On
the
other
hand,
σ̃
also
induces
a
commutative
diagram
of
semi-graphs
∼
Γ
Z
T
−−−−→
Γ
Z
T
σ
⏐
⏐
⏐
⏐
∼
Γ
Y
T
−−−−→
Γ
Y
T
σ
compatible
with
the
respective
natural
actions
of
Gal(Z/Y
),
Gal(Z
σ
/Y
σ
)
[cf.
Proposition
2.3,
(iv);
[SemiAn],
Corollary
3.11].
Thus,
since
the
isomorphism
∼
Γ
Z
T
→
Γ
Z
T
σ
of
dual
semi-graphs
is
compatible
with
the
respective
metric
struc-
∼
tures,
we
conclude
from
Proposition
3.15
again
that
the
isomorphism
Γ
Y
T
→
Γ
Y
T
σ
of
dual
semi-graphs
is
also
compatible
with
the
respective
metric
struc-
tures.
Finally,
it
follows
immediately
from
the
well-known
theory
of
pointed
stable
curves
and
contraction
morphisms
that
arise
from
eliminating
cusps,
as
exposed
in
[Knud]
[cf.
also
Remark
2.1.4],
that
this
implies
that,
if
we
write
Y,
Y
σ
for
the
respective
compactified
stable
models
of
Y
,
Y
σ
over
O
Q
p
,
then
∼
the
isomorphism
Γ
Y
→
Γ
Y
σ
of
dual
semi-graphs
induced
by
σ̃
[cf.
[SemiAn],
Corollary
3.11]
is
compatible
with
the
respective
metric
structures.
Thus,
we
conclude
from
[CbTpIII],
Definition
3.7,
(ii);
[CbTpIII],
Remark
3.19.2,
that
GT
M
=
GT
tp
p
.
This
completes
the
proof
of
Theorem
3.16.
Remark
3.16.1.
Here,
we
recall
that
one
of
the
key
ingredients
in
the
proof
of
[Tsjm],
Corollary
B,
is
the
theory
of
resolution
of
nonsingularities
developed
in
[Lpg1].
As
a
corollary,
we
obtain
the
following
affirmative
answer
to
the
question
posed
in
the
discussion
immediately
preceding
Theorem
E
in
[CbTpIII],
Intro-
duction:
Corollary
3.17
(Commensurable
terminality
of
various
p-adic
versions
of
the
Grothendieck-Teichmüller
group).
We
maintain
the
notation
of
Theorem
3.16.
Then
GT
M
=
GT
p
=
GT
G
=
GT
tp
p
is
commensurably
terminal
M
in
GT,
i.e.,
the
commensurator
C
GT
(GT
)
of
GT
M
in
GT
is
equal
to
GT
M
.
96
Proof.
It
follows
immediately
from
Theorem
3.16,
together
with
[CbTpIII],
The-
orem
E,
that
GT
M
⊆
C
GT
(GT
M
)
⊆
GT
G
=
GT
M
.
Thus,
we
conclude
that
C
GT
(GT
M
)
=
GT
M
,
as
desired.
Proposition
3.18
(Reconstruction
of
the
subset
of
Q
p
-rational
points
def
from
the
p-adic
Grothendieck-Teichmüller
group).
Write
X
=
P
1
Q
\
p
{0,
1,
∞};
Π
tp
X
for
the
Primes-tempered
fundamental
group
of
X.
Then
the
subset
X(Q
p
)
⊆
X(C
p
),
where
we
think
of
“X(C
p
)”
as
the
set
reconstructed
from
Π
tp
X
in
Corollary
3.10,
may
be
reconstructed,
in
a
purely
combinatorial/group-
theoretic
way,
from
the
data
tp
tp
(Π
tp
X
,
GT
p
⊆
Out(Π
X
))
tp
—
consisting
of
the
underlying
topological
group
of
Π
tp
X
and
the
subgroup
GT
p
⊆
tp
Out(Π
X
)
—
as
the
subset
of
elements
fixed
by
some
open
subgroup
of
GT
tp
p
.
Moreover,
this
reconstruction
procedure
is
functorial
with
respect
to
isomor-
phisms
of
topological
groups
for
which
the
induced
isomorphism
on
“Out(−)”
preserves
the
given
subgroup
of
“Out(−)”.
Proof.
First,
we
observe
that
it
follows
immediately
from
the
existence
of
the
natural
homeomorphism
“θ
X
”
of
Proposition
2.3,
(viii),
together
with
the
defi-
tor
”
[cf.
Definition
2.2,
(vi)],
that
the
subset
X(Q
p
)
⊆
X(C
p
)
nition
of
“VE(
X)
is
dense
in
X(C
p
),
and
that
the
natural
action
of
GT
tp
p
on
X(C
p
)
is
via
self-
homeomorphisms
of
X(C
p
)
[cf.
Corollary
3.10
and
its
proof;
Corollary
3.16].
Thus,
since
the
natural
action
of
GT
tp
p
on
X(Q
p
)
factors
through
the
surjection
tp
GT
p
G
Q
p
[cf.
[Tsjm],
Corollary
B,
and
its
proof],
we
conclude
that
the
tp
natural
action
GT
tp
p
on
X(C
p
)
factors
through
this
surjection
GT
p
G
Q
p
,
and
hence
[cf.
[Tate],
§3.3,
Theorem
1]
that
the
subset
X(Q
p
)
⊆
X(C
p
)
may
be
characterized
as
the
subset
of
elements
fixed
by
some
open
subgroup
of
GT
tp
p
.
This
completes
the
proof
of
Proposition
3.18.
Finally,
we
apply
the
theory
of
resolution
of
nonsingularities
and
point-
theoreticity
[cf.,
especially,
Corollary
2.5,
(i);
Corollary
3.10],
together
with
the
theory
of
metric-admissibility
developed
in
[CbTpIII],
§3,
to
construct
certain
arithmetic
cuspidalizations
of
the
[Primes-]tempered
fundamental
groups
of
hy-
perbolic
curves
over
Q
p
equipped
with
“proj-metric
structures”
[cf.
Definition
3.19
below].
Definition
3.19.
Let
Σ
⊆
Primes
be
a
subset
of
cardinality
≥
2
that
con-
tains
p;
K
a
mixed
characteristic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
hyperbolic
curve
over
K.
Write
Π
tp
X
for
the
Σ-tempered
∗
fundamental
group
of
X.
For
each
open
subgroup
Π
⊆
Π
tp
X
of
finite
index,
write
97
•
X
Π
∗
for
the
compactified
stable
model
over
O
K
of
the
hyperbolic
curve
over
K
corresponding
to
the
open
subgroup
Π
∗
⊆
Π
tp
X
;
•
Γ
Π
∗
for
the
dual
graph
associated
to
(X
Π
∗
)
s
;
•
μ
Π
∗
for
the
metric
structure
on
Γ
Π
∗
associated
to
X
Π
∗
,
considered
up
to
multiplication
by
a
constant
∈
Q
×
[cf.
[CbTpIII],
Definition
3.5,
(iii)].
Then
we
shall
refer
to
μ
Π
∗
as
the
proj-metric
structure
on
Γ
Π
∗
.
We
shall
refer
to
the
collection
of
data
of
proj-metric
structures
{μ
Π
∗
}
associated
to
the
charac-
teristic
open
subgroups
{Π
∗
⊆
Π
tp
X
}
of
finite
index
as
the
proj-metric
structure
tp
on
Π
X
.
Theorem
3.20
(Construction
of
certain
arithmetic
cuspidalizations
of
geometric
tempered
fundamental
groups).
Let
K
be
a
mixed
characteris-
tic
complete
discrete
valuation
field
of
residue
characteristic
p;
X
a
hyperbolic
a
universal
pro-Primes
covering
of
X.
Suppose
that
X
sat-
curve
over
K;
X
isfies
Primes-RNS.
Write
Ω
for
the
p-adic
completion
of
K.
For
n
≥
2
an
def
integer,
write
X
n
for
the
n-th
configuration
space
associated
to
X;
Π
1
=
Π
X
;
def
Π
n
=
Π
X
n
;
Π
2/1
for
the
kernel
of
the
natural
surjection
Π
2
Π
1
induced
by
the
first
projection
X
2
→
X,
where
we
regard
Π
2
as
a
quotient
of
Π
n
via
the
projection
X
n
→
X
2
to
the
first
two
factors;
Π
tp
1
for
the
Primes-tempered
fundamental
group
of
X;
def
(Out(Π
n
)
⊇)
Out(Π
n
)
tp
=
Out
gF
(Π
n
)
×
Out(Π
1
)
Out(Π
tp
1
),
def
(Out(Π
n
)
⊇)
Out
gFC
(Π
n
)
=
Out
gF
(Π
n
)
∩
Out
FC
(Π
n
)
(⊆
Out(Π
1
)),
def
(Out(Π
n
)
⊇)
Out
gFC
(Π
n
)
M
=
Out
gF
(Π
n
)
∩
Out
FC
(Π
n
)
M
(⊆
Out(Π
tp
1
)
⊆
Out(Π
1
))
[cf.
[HMM],
Definition
2.1,
(iv);
[CbTpI],
Theorem
A,
(i);
[CbTpIII],
Propo-
sition
3.3,
(iv);
[CbTpIII],
Definition
3.7,
(i),
(ii),
(iii);
[NodNon],
Theorem
B];
VE(Π
tp
Π
tp
1
),
1
(Ω)
and
“
X(Ω)”
for
the
respective
sets
“VE(
X)”
equipped
with
their
natural
actions
tp
by
Aut(Π
1
)
and
Π
1
constructed
in
[the
proof
of
]
Corollary
2.5,
(i),
and
Corol-
tp
lary
3.10
from
the
underlying
topological
group
of
Π
tp
1
.
Let
x̃
∈
VE(Π
1
).
Then
the
following
hold:
(i)
One
may
construct
an
“arithmetic
cuspidalization”
of
Π
tp
1
associated
to
x̃
from
the
data
consisting
of
•
the
topological
group
Π
n
equipped
with
the
quotients
Π
n
Π
2
Π
1
and
a
topology
[i.e.,
the
tempered
topology]
on
the
subquotient
Π
n
Π
2
Π
1
⊇
Π
tp
1
98
in
a
fashion
that
is
functorial
with
respect
to
isomorphisms
of
this
data
[in
the
evident
sense]
as
follows:
Observe
that
the
subgroup
Out(Π
n
)
tp
⊆
Out(Π
n
)
may
be
constructed
from
the
given
data
[cf.
[HMM],
Theorem
A,
(ii)].
Write
tp
tp
out
tp
n
D
x̃
⊆
Π
1
Out(Π
n
)
tp
=
Aut(Π
tp
1
)
×
Out(Π
tp
)
Out(Π
n
)
1
[cf.
[CbTpIII],
Proposition
3.3,
(i),
(ii);
[MT],
Proposition
2.2,
(ii)]
for
the
stabilizer
subgroup
of
x̃.
Note
that
there
exists
a
natural
exact
sequence
[that
may
be
constructed
from
the
given
data]
out
out
tp
tp
1
−→
Π
2/1
−→
(Π
2
×
Π
1
Π
tp
−→
Π
tp
−→
1.
1
)
Out(Π
n
)
1
Out(Π
n
)
Thus,
by
pulling-back
the
above
exact
sequence
via
the
inclusion
n
D
x̃
tp
⊆
out
tp
Π
tp
1
Out(Π
n
)
,
we
obtain
an
exact
sequence
out
1
−→
Π
2/1
−→
Π
2/1
n
D
x̃
tp
−→
n
D
x̃
tp
−→
1.
out
We
shall
refer
to
Π
2/1
n
D
x̃
tp
as
the
[n-th]
arithmetic
cuspidalization
of
Π
tp
1
associated
to
x.
(ii)
Write
n-alg
Π
tp
⊆
Π
tp
1
(Ω)
1
(Ω)
for
the
subset
of
elements
ξ
∈
Π
tp
1
(Ω)
whose
Π
1
-orbit
Π
1
·
ξ
is
stabilized
by
some
open
subgroup
of
Out
gFC
(Π
n
)
M
(⊆
Out(Π
tp
1
))
[cf.
Remark
3.20.1
n-alg
⊆
Π
tp
below].
Suppose
that
x̃
arises
from
an
element
∈
Π
tp
1
(Ω)
1
(Ω),
which,
by
a
slight
abuse
of
notation,
we
shall
also
denote
by
x̃.
Write
def
X
x
=
X
Ω
\
{x},
where
x
∈
X(Ω)
denotes
the
element
determined
by
x̃.
Then
the
[Primes-]tempered
fundamental
group
Π
tp
X
x
(⊆
Π
2/1
)
of
X
x
[where
we
identify
Π
2/1
with
Π
X
x
],
together
with
the
proj-metric
structure
on
Π
tp
X
x
,
may
be
reconstructed,
in
a
purely
combinatorial/group-
theoretic
way,
from
the
following
data
•
the
topological
group
Π
n
equipped
with
the
quotients
Π
n
Π
2
Π
1
and
a
topology
[i.e.,
the
tempered
topology]
and
proj-metric
structure
on
the
subquotient
Π
n
Π
2
Π
1
⊇
Π
tp
1
;
•
the
subgroup
Out
gFC
(Π
n
)
⊆
Out(Π
n
)
[cf.
Remark
3.20.2
below]
in
a
fashion
that
is
functorial
with
respect
to
isomorphisms
of
this
data
[in
the
evident
sense].
99
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(ii).
Write
M
def
=
n
D
x̃
out
gFC
Π
tp
(Π
n
)
M
1
Out
∩
n
D
x̃
tp
out
tp
(⊆
Π
tp
1
Out(Π
n
)
).
In
particular,
it
follows
immediately
from
assertion
(i),
together
with
the
various
definitions
involved,
that
one
may
construct,
from
the
given
data,
n
D
x̃
M
,
together
with
the
natural
outer
action
of
n
D
x̃
M
on
Π
2/1
.
Let
Π
†
2/1
⊆
Π
2/1
be
an
open
subgroup
that
is
normal
in
Π
2
;
l
a
prime
number
=
p.
Write
K
⊆
K
tm
(⊆
K)
def
tm
/K
ur
);
Π
2/1
Π
∗
2/1
for
for
the
maximal
tame
extension
of
K;
G
tm
K
=
Gal(K
the
maximal
almost
pro-l
quotient
associated
to
[i.e.,
“with
respect
to”]
Π
†
2/1
⊆
Π
2/1
[cf.
[CbTpIII],
Definition
1.1].
Let
us
assume
further
that
the
quotient
Π
2
Π
2
/Ker(Π
2/1
Π
∗
2/1
)
is
F-characteristic
[cf.
[CbTpIII],
Definition
2.1,
(iii)].
Note
that
this
implies
that,
relative
to
the
identification
of
Π
2/1
with
Π
X
x
[cf.
the
statement
of
assertion
(ii)],
the
natural
Π
2/1
-outer
action
of
G
K
on
Π
2
[which
is
well-defined
after
possibly
replacing
K
by
a
suitable
finite
extension
∗
field
of
K]
induces
a
natural
outer
action
of
G
K
on
Π
2/1
.
Thus,
in
order
to
complete
the
proof
of
assertion
(ii),
it
suffices,
in
light
of
the
argument
given
in
the
proof
of
[CbTpIII],
Theorem
3.9
[cf.,
especially,
the
equivalence
stated
in
the
final
display
of
the
proof
of
[CbTpIII],
Theorem
3.9],
to
reconstruct,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
the
image
of
G
tm
K
in
Out(Π
∗
2/1
)
via
the
natural
outer
representation.
Next,
we
verify
the
following
assertion:
→)
Y
→
X
be
a
[connected]
finite
étale
Galois
Claim
3.20.A:
Let
(
X
covering
over
K.
Then
there
exist
a
compactified
semistable
model
→
Y
of
Y
over
O
K
and
a
[connected]
finite
étale
Galois
covering
(
X
)
Z
→
X
over
K
that
dominates
Y
→
X
and
satisfies
the
following
conditions:
•
The
compactified
stable
model
Z
of
Z
over
O
K
dominates
Y.
•
The
component
of
the
VE-chain
x̃
corresponding
to
Z
is
an
irreducible
component
of
Z
s
that
maps
to
a
smooth
closed
point
∈
Y
s
that
is
not
a
cusp.
Indeed,
since
X
satisfies
Primes-RNS,
Claim
3.20.A
follows
immediately
from
the
fact
that
the
VE-chain
x̃
arises
from
an
element
∈
Π
tp
1
(Ω)
that
is
of
type
1,
hence
weakly
verticial
[cf.
Proposition
2.4,
(v);
Remark
3.1.1;
Proposition
3.3,
(ii);
Proposition
3.4,
(iii);
the
theory
of
pointed
stable
curves,
as
exposed
in
[Knud]].
Next,
we
observe
that
it
follows
from
[CbTpIII],
Proposition
2.3,
(ii)
[cf.
conditions
(a),
(b),
(c)
below];
[CbTpIII],
Corollary
2.10
[cf.
condition
(c)
be-
low],
together
with
Claim
3.20.A
[cf.
condition
(b)
below],
that,
after
possibly
replacing
Π
†
2/1
⊆
Π
2/1
by
a
smaller
open
subgroup
that
satisfies
the
same
con-
ditions
as
Π
†
2/1
,
there
exist
F-characteristic
SA-maximal
almost
pro-l
quotients
Π
2
Π
∗
2
,
Π
2
Π
∗∗
2
satisfying
the
following
conditions:
100
(a)
The
F-characteristic
SA-maximal
almost
pro-l
quotient
Π
2
Π
∗∗
2
dom-
inates
the
F-characteristic
SA-maximal
almost
pro-l
quotient
Π
2
Π
∗
2
.
In
particular,
we
obtain
a
commutative
diagram
of
profinite
groups
∗∗
∗∗
1
−−−−→
Π
∗∗
2/1
−−−−→
Π
2
−−−−→
Π
1
−−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
1
−−−−→
Π
∗
2/1
−−−−→
Π
∗
2
−−−−→
Π
∗
1
−−−−→
1,
∗
∗∗
where
the
quotients
Π
∗
1
,
Π
∗∗
1
of
Π
1
induced
by
Π
2
,
Π
2
are
the
center-free
[cf.
[CbTpIII],
Proposition
1.7,
(i)]
maximal
almost
pro-l
quotients
of
Π
1
associated
to
normal
open
subgroups
of
Π
1
;
the
quotients
Π
∗
2/1
,
Π
∗∗
2/1
of
Π
2/1
induced
by
Π
∗
2
and
Π
∗∗
are
the
center-free
[cf.
[CbTpIII],
Proposition
2
1.7,
(i)]
maximal
almost
pro-l
quotients
of
Π
2/1
associated
to
normal
open
subgroups
of
Π
2/1
;
the
vertical
arrows
denote
surjective
homomorphisms.
(b)
Fix
a
normal
open
subgroup
Π
Y
⊆
Π
1
whose
associated
maximal
almost
pro-l
quotient
coincides
with
Π
1
Π
∗
1
.
Then
there
exists
a
normal
open
subgroup
Π
Z
⊆
Π
1
such
that:
•
It
holds
that
Π
Z
⊆
Π
Y
.
[In
particular,
the
maximal
almost
pro-l
quotient
associated
to
the
normal
open
subgroup
Π
Z
⊆
Π
1
dominates
the
maximal
almost
pro-l
quotient
Π
1
Π
∗
1
.]
•
The
maximal
almost
pro-l
quotient
Π
1
Π
∗∗
1
dominates
the
maximal
almost
pro-l
quotient
associated
to
the
normal
open
subgroup
Π
Z
⊆
Π
1
.
→)
Y
→
X,
(
X
→)
Z
→
X
for
the
respective
[connected]
•
Write
(
X
finite
étale
Galois
coverings
over
K
associated
to
the
normal
open
subgroups
Π
Y
⊆
Π
1
,
Π
Z
⊆
Π
1
.
Then
there
exists
a
compactified
semistable
model
Y
of
Y
over
O
K
such
that
the
compactified
stable
model
Z
of
Z
over
O
K
dominates
Y,
and,
moreover,
the
component
of
the
VE-chain
x̃
corresponding
to
Z
is
an
irreducible
component
of
Z
s
that
maps
to
a
smooth
closed
point
∈
Y
s
that
is
not
a
cusp.
FC
∗
∗∗
(Π
∗∗
(c)
Every
element
∈
Out
FC
(Π
∗∗
2
Π
2
)
∩
Ker(Out
2
)
→
Out(Π
1
))
[cf.
[CbTpIII],
Definition
2.1,
(viii)]
induces
the
trivial
outer
automorphism
of
Π
∗
2
.
Write
D
x̃
∗∗
out
out
for
the
image
of
n
D
x̃
M
(⊆
Π
1
Out
gFC
(Π
n
)
⊆
Π
1
Out
gFC
(Π
2
))
[where
the
second
inclusion
follows
from
[NodNon],
Theorem
B]
via
the
natural
homomor-
out
out
FC
phism
Π
1
Out
gFC
(Π
2
)
→
Π
∗∗
(Π
∗∗
1
Out
2
);
out
FC
∗
∗∗
ρ
∗∗
:
D
x̃
∗∗
⊆
Π
∗∗
(Π
∗∗
1
Out
2
Π
2
)
−→
Out(Π
1
)
101
for
the
natural
composite
homomorphism.
Next,
we
verify
the
following
assertion:
Claim
3.20.B:
There
exists
an
open
subgroup
D
∗∗
⊆
D
x̃
∗∗
such
that
every
element
∈
D
∗∗
∩
Ker(ρ
∗∗
)
induces
the
trivial
outer
action
on
Π
∗
2/1
.
Indeed,
observe
that
since
Π
∗∗
1
is
center-free
[cf.
condition
(a)],
the
natural
homomorphism
out
∗∗
∗∗
D
x̃
∗∗
−→
Π
∗∗
1
Out(Π
1
)
=
Aut(Π
1
)
induces
a
natural
homomorphism
φ
:
Ker(ρ
∗∗
)
−→
Π
∗∗
1
.
Thus,
it
follows
immediately
from
the
final
portion
of
condition
(b)
[cf.
also
the
proof
of
Claim
3.20.A;
our
assumption
that
X
satisfies
Primes-RNS;
the
portion
of
Proposition
3.5,
(i),
concerning
the
weakly
verticial
case],
together
with
the
various
definitions
involved
[cf.,
especially,
the
definition
of
D
x̃
∗∗
],
that
the
image
of
the
natural
composite
homomorphism
φ
∗
Ker(ρ
∗∗
)
−→
Π
∗∗
1
Π
1
is
finite.
In
particular,
there
exists
an
open
subgroup
D
∗∗
⊆
D
x̃
∗∗
such
that
every
element
∈
D
∗∗
∩
Ker(ρ
∗∗
)
induces
the
trivial
automorphism
of
Π
∗
1
.
On
the
other
hand,
this
implies,
in
light
of
condition
(c),
that
every
element
∈
D
∗∗
∩
Ker(ρ
∗∗
)
induces
the
trivial
outer
automorphism
of
Π
∗
2
.
Thus,
since
Π
∗
1
is
center-free
[cf.
condition
(a)],
we
conclude
that
every
element
∈
D
∗∗
∩
Ker(ρ
∗∗
)
induces
the
trivial
outer
automorphism
of
Π
∗
2/1
.
This
completes
the
proof
of
Claim
3.20.B.
Next,
let
us
observe
that,
by
applying
the
argument
given
in
the
proof
of
Theorem
3.9
[cf.,
especially,
the
equivalence
stated
in
the
final
display
of
the
proof
of
[CbTpIII],
Theorem
3.9],
together
with
condition
(a),
we
conclude
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
one
tm
may
reconstruct,
from
the
proj-metric
structure
on
Π
tp
1
,
the
image
I
of
G
K
in
Out(Π
∗∗
)
via
the
natural
outer
representation.
On
the
other
hand,
observe
1
that
ρ
∗∗
(D
∗∗
)
contains
an
open
subgroup
of
I
[cf.
Remark
3.20.1
below].
Thus,
since
D
∗∗
∩
Ker(ρ
∗∗
)
induces
the
trivial
outer
action
on
Π
∗
2/1
[cf.
Claim
3.20.B],
by
considering
the
outer
action
of
D
∗∗
∩
(ρ
∗∗
)
−1
(I)
on
Π
∗
2/1
,
we
conclude
that,
after
possibly
replacing
K
by
a
suitable
finite
extension
field
of
K,
one
may
∗
reconstruct
the
image
of
G
tm
K
in
Out(Π
2/1
),
as
desired.
This
completes
the
proof
of
assertion
(ii),
hence
of
Theorem
3.20.
102
Remark
3.20.1.
Suppose
that
we
are
in
the
situation
of
Theorem
3.20,
(ii).
Then
it
follows
immediately
from
the
definitions
[cf.
also
the
natural
homomorphism
G
K
→
Out
gFC
(Π
n
)
M
[which
is
well-defined
after
possibly
replacing
K
by
a
suit-
able
finite
extension
field
of
K];
the
proof
of
Theorem
3.11,
(ii)]
that
the
subset
tp
tp
n-alg
Π
tp
⊆
Π
tp
1
(Ω)
1
(Ω)
is
contained
in
the
subset
Π
1
(K)
⊆
Π
1
(Ω)
correspond-
constructed
in
Corollary
3.10.
ing
to
the
K-rational
points
of
the
set
“
X(Ω)”
Moreover,
it
follows
immediately
from
Proposition
3.18
[cf.
also
Theorem
3.16;
[HMM],
Corollaries
B,
C]
that
the
inclusion
n-alg
⊆
Π
tp
Π
tp
1
(Ω)
1
(K)
is
in
fact
an
equality
in
the
case
where
X
=
P
1
Q
\
{0,
1,
∞}.
It
is
not
clear
to
p
the
authors,
however,
at
the
time
of
writing
of
the
present
paper
whether
or
not
the
inclusion
of
the
above
display
is
an
equality
in
general.
Remark
3.20.2.
Suppose
that
we
are
in
the
situation
of
Theorem
3.20,
(ii).
Then
we
observe
that
the
data
“Out
gFC
(Π
n
)
⊆
Out(Π
n
)”
may
be
omitted
from
the
list
of
data
in
the
statement
of
Theorem
3.20,
(ii),
in
either
of
the
following
situations
[cf.
[CbTpII],
Theorem
A,
(ii);
[HMM],
Theorem
A,
(ii);
[HMT],
Corollary
2.2]:
•
X
is
of
genus
0.
•
X
is
affine,
and
n
≥
3.
•
X
is
proper,
and
n
≥
4.
It
is
not
clear
to
the
authors,
however,
at
the
time
of
writing
of
the
present
paper
whether
or
not
this
data
“Out
gFC
(Π
n
)
⊆
Out(Π
n
)”
may
be
omitted
in
general.
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106
Updated
versions
of
[HMT],
[CbTpIII],
[CbTpIV]
may
be
found
at
the
following
URL:
http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
(Shinichi
Mochizuki)
Research
Institute
for
Mathematical
Sciences,
Kyoto
University,
Kyoto
606-8502,
Japan
Email
address:
motizuki@kurims.kyoto-u.ac.jp
(Shota
Tsujimura)
Research
Institute
for
Mathematical
Sciences,
Kyoto
Uni-
versity,
Kyoto
606-8502,
Japan
Email
address:
stsuji@kurims.kyoto-u.ac.jp
107